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Enhancement of Charged Particle
Emission from a Plasma Focus Device
A Thesis Submitted to the
College of Graduate and Postdoctoral Studies
In Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy in
Physics in the Department of Physics and
Engineering Physics
University of Saskatchewan
Saskatoon, Saskatchewan, Canada
By
Reza Alibazi Behbahani
©Copyright Reza Behbahani, August 2017. All rights reserved.
University of Saskatchewan
i
Permission to Use
In presenting this thesis in partial fulfillment of the requirements for a Postgraduate degree from
the University of Saskatchewan, I agree that the Libraries of this University may make it freely
available for inspection. I further agree that permission for copying of this thesis/dissertation in
any manner, in whole or in part, for scholarly purposes may be granted by the professor or
professors who supervised my thesis work or, in their absence, by the Head of the Department or
the Dean of the College in which my thesis work was done. It is understood that any copying or
publication or use of this thesis/dissertation or parts thereof for financial gain shall not be allowed
without my written permission. It is also understood that due recognition shall be given to me and
to the University of Saskatchewan in any scholarly use which may be made of any material in my
thesis.
Head of the Department of Physics and Engineering Physics
163-116 Science Place
University of Saskatchewan
Saskatoon, Saskatchewan S7N 5E2 Canada
ii
Abstract
Design and optimization of a low energy plasma focus for enhancing charged particle emission
and x-ray radiation from the plasma focus device have been investigated. The design concept and
the technical details are described. Design and fabrication of the required diagnostic instruments
used to monitor the emitted charged particles and x-rays radiations are presented.
The effects of working gas on pinching regimes of the device and generation of charged particle
and x-ray radiation have been investigated. Hydrogen, nitrogen, and argon have been used as
working gases. The energy spectra of the ions generated in three operating gases depend on the
capacitor bank voltage and the effective charge of the ions. The duration of the current drop and
the injected energy into the plasma increases significantly in the high-z gases such as argon. Based
on discharge circuit analysis of the device it has been found that the plasma resistance reaches 0.2
ohm during the post pinch phase in argon plasma and this anomalous resistance causes significant
energy consumption during this phase.
It has also been found that the anomalous Joule heating during the post pinch phase enhances the
production rate of the runaway charged particles due to reduced Dreicer field. It has been found
that the runaway electron generation conditions in tokamaks and plasma focus devices are
consistent in terms of the ratio of the electric field across the plasma to the electron plasma density.
To enhance the efficiency of plasma focus device as a charged particle source, a new configuration
of a dense plasma focus device with three electrodes has been designed and tested. The preliminary
experimental results have demonstrated that the new device can produce several focusing events
in a plasma focus device. The ion beam emission and hard x-ray radiation pulses last longer than
that is expected in a conventional plasma focus and their intensities are also higher.
iii
Acknowledgements
I would like to extend my gratitude first and foremost to my supervisor Dr. Chijin Xiao for
his mentorship over the course of my Ph.D. study. His insight has helped me through extremely
tough times during my research. He is the one that was always with me when I had difficulties.
I would also like to extend my sincere appreciation to Dr. Akira Hirose, the director of the Plasma
Physics Laboratory, for his excellent support and guidance. I would also like to thank Mr. David
McColl and all my fellow graduate students.
This research was supported by the grants from the Natural Sciences and Engineering Research
Council of Canada (NSERC), and the Sylvia Fedoruk Canadian Centre for Nuclear Innovation.
The partial financial aid from the University of Saskatchewan and Saskatchewan Innovation
Opportunity Scholarship is also much appreciated.
iv
This is dedicated to my parents
v
Table of Contents
PERMISSION TO USE ………….……………………………………...………………………...I
ABSTRACT....................................................................................................................................II
ACKNOWLEDGEMENTS ………………………..…………..………………………………..III
DEDICATION……………………………………………………………………………….…..IV
TABLE OF TABLE ……………..………………………………………..……………….….….V
TABLE OF FIGURES…………………………………………………………………..…......VIII
LIST OF TABLES……………………………….…….……………………………..………...XII
LIST OF ABBREVIATIONS………………………….…………………………………..…..XIII
1. INTRODUCTION…………………………………..…………………………………………..1
1.1 Magnetic Pinch ........................................................................................................................ 1
1.2 Dense Plasma Focus ................................................................................................................ 4
1.3 Research History on Dense Plasma Focus ............................................................................ 9
1.3.1 Plasma Dynamics ................................................................................................................. 9
1.3.2 Charged Particles Emission ............................................................................................... 11
1.3.3 X-ray Radiation .................................................................................................................. 14
1.3.4 Neutron Emission ............................................................................................................... 15
1.4 Dense Plasma Focus Applications ....................................................................................... 17
1.4.1 Short-lived Radioisotopes Production……………………………………………………..17
vi
1.4.2 Thin Film Deposition and Ion Implantation .................................................................... 20
1.4.3 Detection of Illicit Materials and Explosives .................................................................. 22
1.5 Motivation of the Thesis ....................................................................................................... 24
1.6 Thesis Outline ........................................................................................................................ 25
2. PHYSICS OF Z-PINCH……………………………………………………………………….27
2.1 Introduction ........................................................................................................................... 27
2.2 Lee Model .............................................................................................................................. 32
2.2.1 Axial Phase ......................................................................................................................... 32
2.2.2 Inward Shockwave Phase .................................................................................................. 36
2.2.3 Reflected Shockwave Phase .............................................................................................. 40
2.2.4 Slow Compression Phase................................................................................................... 42
2.2.5 Instability Phase ................................................................................................................. 44
2.3 Lee Code ................................................................................................................................ 45
3. DPF UOFS-1……………………………………………………………...……………..…….46
3.1 Introduction ........................................................................................................................... 46
3.2 Optimization of DPF UofS-I as an Ion Source ................................................................... 47
3.3 Experimental Setup of DPF UofS-I ..................................................................................... 52
4. DPF UOFS-1 DIAGNOSTICS ................................................................................................. 60
4.1 Introduction ............................................................................................................................. 60
4.1.1 Discharge Current and Anode Voltage Probes .................................................................... 61
vii
4.1.2 Faraday Cup ......................................................................................................................... 63
4.1.3 Electron Beam Measurement ............................................................................................... 67
4.1.4 Hard X-ray Measurement .................................................................................................... 70
4.1.5 Soft X-ray Measurement ...................................................................................................... 70
4.1.5 Signal Recording System ..................................................................................................... 75
5. EXPERIMENTAL RESULTS AND DISCUSSION ............................................................... 77
5.1 Introduction ............................................................................................................................. 76
5.2 Charged Particle Emission in Light and Heavy Gases ........................................................... 77
5.3 Anomalous Resistance and Injected Energy into Plasma ....................................................... 87
5.4 Plasma Heating and the Emission of Runaway Charged Particle........................................... 90
5.5 DPF Device with Semi-Active Electrodes (DPF-SAE) .......................................................... 97
5.5.1 Configuration of DPF-SAE.................................................................................................. 98
5.5.2 Discharge Circuit Analyses of DPF-SAE .......................................................................... 106
5.5.3 Charged Particle Emission and X-ray Radiation from the DPF-SAE Device ................... 109
5.5.4 Comparison of Ion and X-ray Radiation between DPF-SAE and DPF UofS-I Devices.. . 111
6. SUMMARY . .......................................................................................................................... 114
6.1 Summary ............................................................................................................................... 114
6.2 Suggested Future Research ................................................................................................... 117
REFERENCES ........................................................................................................................... 119
APPENDIX ................................................................................................................................. 127
viii
Table of Figures
Fig. 1.1: A Z-pinch in cylindrical coordinate ..................................................................................3
Fig. 1.2: Schematic of theta pinch configuration .............................................................................3
Fig. 1.3: The configuration of X-pinch. ..........................................................................................4
Fig. 1.4: Three positions of the current sheath in plasma focus device. ..........................................7
Fig. 1.5: Typical discharge current and tube voltage traces during focusing of plasma ……….....8
Fig. 1.6 Filippov-type dense plasma focus ......................................................................................9
Fig. 1.7: The directions of charged particle beam emission from DPF device .............................12
Fig. 1.8: The bombardment of target by ion beam generated by plasma ....................................18
Fig. 1. 9: Schematic arrangement for thin film deposition in a plasma focus ..............................21
Fig. 1.10: Schematic arrangement of illicit and explosive materials detection by a DPF ............23
Fig. 2.1: Macro instabilities such as m=0 and m=1 through the plasma column…………..…….30
Fig. 2.2 Formation of current sheath in plasma focus…..……………………………………......33
Fig. 2.3: The axial phase in a Mather type plasma focus ..............................................................33
Fig. 2.4: The inward radial shock wave in final stage of plasma focus .........................................36
Fig. 2.5: The reflected shockwave phase in plasma focus ............................................................41
Fig. 2.6: The simulated discharge current and voltage by Lee code in three phases ....................45
Fig. 3.1: The discharge current approximately140 kA and tube voltage reaches up to 20 kV..….49
ix
Fig.3.2: The peak axial velocity about 10 cm/µsec under the operation conditions…..………....50
Fig. 3.3: The simulated plasma temperature and ion density in DPF UofS -I ...............................51
Fig. 3.4: The DPF UofS-I built in the University of Saskatchewan ............................................53
Fig. 3.5: The assembly of electrodes and detectors on the vacuum chamber ..............................54
Fig. 3.6: The schematic of spark gap used in DPF UofS-I ...........................................................55
Fig. 3.7: Faraday cage used for EMP shielding .............................................................................56
Fig. 3.8: The focusing effect of DPF UofS-I operating in argon gas ............................................59
Fig. 4.1: Rogowski coil equivalent circuit .....................................................................................61
Fig. 4.2: The current probe and high voltage probe were used in DPF UofS-I ...........................63
Fig. 4.3: The Faraday cup assembly used in DPF-I. .....................................................................64
Fig. 4.4: The configuration of Faraday cup was used to monitor generated ion beam ………….65
Fig. 4.5: Ion beam current (IB) and discharge current (I) signals .................................................65
Fig. 4.6: A typical signal of electron beam (I-E-beam) generated from DPF UofS-I .................69
Fig. 4.7: The typical signals of hard x-ray from DPF UofS-I .....................................................70
Fig. 4.8: The biasing circuit of BPX-65 diode ..............................................................................71
Fig. 4.9: The sensitivity of BPX-65 respect to energy spectrum of x-ray ....................................72
Fig. 4.10: The two soft x-ray detectors were used in DPF UofS-I ...............................................74
Fig. 4.11: The soft x-ray signal from DPF UofS-I ........................................................................74
Fig. 5.1: The different regimes of focusing in argon and hydrogen gasses ..................................78
x
Fig. 5.2: Discharge current and tube voltage in argon gas during the compression phase ............80
Fig. 5.3: Change in tube inductance in hydrogen gas ....................................................................81
Fig. 5.4: Simulated radial and axial trajectories of plasma in hydrogen (a) and argon (b)…….....83
Fig. 5.5: Power into discharge tube in argon gas during the compression phase. ........................84
Fig. 5.6: Anode voltage, hard x-ray and ion beam signals in hydrogen gas .................................85
Fig. 5.7: Energy of proton generated from DPF UofS-I in maximum charging voltage……….. 86
Fig. 5.8: The simulated ion current density and the measured one in DPF UofS-I ....................87
Fig. 5.9: The tube voltage, discharge current, and hard x-ray signals .......................................94
Fig. 5.10: Soft x-ray signal from DPF UofS-I ..............................................................................95
Fig. 5.11: Rising time and falling time of x-ray signal .................................................................96
Fig. 5.12: Configuration of DPF-SAE ..........................................................................................99
Fig. 5.13: The equivalent circuit of SAE type plasma focus ......................................................100
Fig. 5.14: Experimental set-up of electrodes in SAE type plasma focus ....................................101
Fig. 5.15: Discharge current and tube voltage based on Lee mode for DPF-I and DPF-II ........102
Fig. 5.16: Positions of the current probe and two high voltage probes ......................................104
Fig. 5.17: Discharge current and tube voltage show two focusing at the same time ……..…....105
Fig. 5.18: Discharge current and tube voltages show three focusing events…...………………106
Fig. 5.19: The measured tube voltage and discharge current in DPF-SAE ................................108
Fig. 5.20: The voltage between electrodes number 1 and 2 and V1 - V2 .....................................108
Fig. 5.21: Two periods of ion beam and hard x-ray emissions for DPF-SAE device………….110
xi
Fig. 5.22: Three periods of ion emissions and hard x-ray radiations in DPF-SAE device …….110
Fig. 5.23: Comparison of ion beams garnered from DPF-SAE and DPF-UofS-I devices ..........112
Fig. 5.24: The comparison between hard x-ray signals in DPF-SAE and DPF-UofS-I device ...113
xii
List of Tables
Table 3.1: The parameters of DPF UofS-I……………………………………………………….52
Table 3.2: The operating conditions of DPF UofS-I .....................................................................58
Table 5.1: The features of the experimental observations for plasma foci the operating gasses. ..79
Table 5.2: Parameters predicted by Lee model for SAE type plasma focus. …………………..103
xiii
List of Abbreviations
DPF Dense plasma focus
DPF-SAE Dense plasma focus with semi-active electrodes
EMP Electromagnetic pulse
FWHM Full width half maximum
H-X Hard x-ray
IB Ion bean current
LHDI Low hybrid drift instability
MCF Magnetic confinement fusion
MP Magnetic piston
ICF Inertial Confinement Fusion
SEE Secondary electron emission mode
SLR Short-lived radioisotope
SW Shock wave
1
Chapter 1
INTRODUCTION
1.1 Magnetic Pinch
A pinch is one of the first configurations proposed for producing fusion energy by
mankind. Z-pinch, theta pinch, and X-pinch are three different types of pinches to compress plasma
by magnetic pressure. A pinch device is a source of intense energetic x-rays. In a pinch device, the
compressed plasma lasts for tens of nanoseconds and instabilities tend to terminate the compressed
plasma [1].
A zeta pinch (Z-pinch) is one of the simplest ways to produce hot and dense plasma in a
laboratory. A Z-pinch is produced by an electrical discharge current in the axial direction (Z axis)
as shown in Fig. 1.1. The induced magnetic field in the azimuthal direction exerts an external
magnetic pressure which compresses the plasma radially inwards. During the compression phase
the plasma density and temperature, and thus the internal pressure, increases. Since the plasma is
highly conductive with a low resistivity, the penetration of magnetic field occurs in a time much
longer than the compression time due to the skin effects. The compression dynamics depends on
the difference between the internal thermal pressure and the external magnetic field. In a normal
Z-pinch device the total time duration of plasma compression is of the order hundreds of
nanoseconds. Therefore, it is necessary to achieve the peak of the discharge current during a very
short time.
2
A high current pulse produced by discharging a low inductance capacitor bank is often
limited to a time duration of several microseconds which is still not short enough for compression
of plasma within hundreds of nanoseconds. To produce a pulse of electrical current with a suitable
time duration the primary electrical pulse from a capacitor bank is transferred to a pulse forming
line (PFL). The PFL would produce an electric current pulse with time duration of hundreds of
nanoseconds and deliver it to the discharge electrodes. Generation of such a pulse of electricity
during a brief time involves expensive equipment and complicated technologies. Usually, a
significant amount of energy could be lost to the PFL.
In a Z-pinch, the magnetic field of discharge current compresses the plasma by the J×B
force in the radial direction, where J is the axial current through the plasma and B the azimuthal
magnetic field produced by the current as illustrated in Fig. 1.1. The name Z-pinch is based on the
direction of the electrical current, which is in the Z-direction in the cylindrical coordinate system.
In Z-pinch devices, the number density of particles reaches up to 1024 m-3, and the plasma
temperature can reach up to 1 keV within hundreds of nanoseconds. This condition is suitable for
D-D and D-T fusion reactions.
A theta pinch is another method to produce a pinching plasma [1]. In a theta pinch, the current
is in the theta direction and the produced magnetic field is in the Z-direction as depicted in
Fig. 1.2. During the compression phase in a theta pinch, the magnetic field inside the plasma is
frozen inside of the plasma due to the high plasma conductivity.
3
Fig. 1.1 A Z-pinch in cylindrical coordinates.
Fig. 1.2 Schematic of theta pinch configuration.
An X-pinch is another configuration for producing a dense and hot plasma by a magnetic
force [1]. In an X-pinch, the current passes through an x-shaped wires. The plasma is then focused
in a cross point of the wires due to the magnetic pressure as shown in Fig. 1.3.
Magnetic field line
in 𝜽 direction
Z direction
Current in primary loop
Plasma
Plasma Current
4
Fig. 1.3: The configuration of an X-pinch device.
1.2 Dense Plasma Focus
The high costs for making a PFL and its complicated process in a Z-pinch machine leads
to the invention of the plasma focus device. Dense plasma focus (DPF) devices have been studied
in 1961 by Filippov [2], and in 1965 by Mather [3]. These studies were carried out in Russia and
United States independently, and led to two different configurations of the plasma focus.
A DPF device consists of a fast capacitor bank, a spark-gap switch, two coaxial electrodes,
an insulator, a vacuum chamber, a vacuum pump, and inlet and outlet gas lines. The electrodes can
be made of copper, brass, or tungsten. Pyrex and quartz can be used to insulate the anode and
cathode. The charging voltage of the capacitor can range between 10 to 50 kV. The discharge
current in a DPF device can reach up to 100 kA in a low energy device (with the stored energy
E ≤ 3 kJ), and 1 MA in a medium energy device (E ≤ 100 kJ), and can reach up to tens of MA in
a high energy device (E ≈1 MJ). In a DPF device, the two coaxial electrodes are insulated by an
C 𝐁𝟐
𝟐µ = Magnetic pressure
Current
S
5
insulator tube (glass or ceramic) and connected to a charged capacitor bank through a switch. The
electrodes are placed in a vacuum chamber filled with the low-pressure gas (0.1 to 20 Torr). When
the switch is triggered, the discharge between the two electrodes breaks down the gas and produces
a current sheath next to insulator. It should be noted that the electrodes and insulator, together,
shape the initial discharge current in the downward axial direction as shown in Fig. 1.4. The
current layer at this stage is accelerated by the J × B force initially in the radial direction and then
also in the axial direction to form an umbrella-shaped current sheath moving axially forward, as
can be seen in Fig. 1.4. Position 1 shows the break-down phase, position 2 shows the axial phase,
and position 3 shows the compression phase. The break-down and axial phases are the two phases
that cause a reasonable delay times in getting the compression of plasma at the peak of the
discharge current.
At the compression phase, a hot and dense plasma column is formed on the top of the anode
and extends in the axial direction. At this stage, charged particle beams and x-rays are generated
by the compressed plasma that can last for tens of nanoseconds. The hot and densely-focused
plasma lasts for a brief period of time (on the order of tens of nanoseconds). The temperature of
the plasma can reach up to 1 keV, and the plasma density can reach up to 1025 m-3. The hot and
dense plasma generates neutrons when Deuterium or Tritium gases are used. The occurrence of
instabilities in a plasma causes a disruption that can induce a huge electric field across the plasma
in the axial direction. The charged particles can be accelerated by electric fields of up to MeV per
meter. The generated electron beams hit the electrodes and generate hard x-rays. Plasma focus
devices may be operated in different gases with low and high atomic numbers, such as hydrogen,
nitrogen, and argon.
6
In all plasma focus devices with different stored energies, the energy density of the
pinching plasma is, 10𝐸
(𝜋𝑎2)𝑎
J
cm3 where a is the anode radius and E the stored energy in the
capacitor bank, is on the same order of magnitude. It should be noted energy density is estimated
based on the ratio of stored energy in capacitor bank and the volume of plasma column
(radius ≈ 0.1a and plasma column length ≈ a). Plasma temperature and electron density are similar
in all devices operating with low and high stored energies. It should be noted that the plasma focus
with a higher stored energy can produce a greater volume of compressed plasma that lasts for a
longer time compared to a low energy device [4]. The drive factor, 𝐼
𝑎√𝑝 (
KA
cm√mbar), where I is the
peak discharge current, a anode radius, and p the operating pressure, is another important
parameter which remains almost in the same range in all DPF devices [4]. In all plasma focus
devices, the energy density is about 1-10 × 1010 J
cm3 and the drive parameter is about
(60 – 120) KA
cm√mbar [4]. Low-Z and High-Z gases such as hydrogen, helium, nitrogen, argon, and
xenon, have been used as working gases for non-fusion applications of plasma focus. For fusion
studies, deuterium and tritium have been used as working gases.
When the plasma is compressed, it will increase in both density and temperature. A
considerable number of neutrons will be emitted from fusion reaction (if Deuterium gas is used).
In addition, the intense radiation of x-rays and a significant number of energetic charged particles
will also be emitted from a plasma focus with low and high stored energies. The low operating
costs of DPF makes this device a strong candidate for industrial applications, for instance, as a
charged particle source. Lithography and microlithography [5], fast x-ray imaging [6], material
7
processing [7], thin film deposition [8], and short-lived isotope production [9] are some of the
industrial applications of the plasma focus to be introduced in the following sections.
Fig. 1.4: Three positions of the current sheath in plasma focus device. “1” denotes the
breakdown phase, “2” axial phase, and “3” radial phase in plasma focus. Red line
shows the plasma layer.
The m=0 and m=1 instabilities commonly seen in DPF plasma induce huge voltages
across the plasma due to the rapid change in plasma inductance [10]. Microinstabilities and
turbulences in plasma can also significantly increase the plasma resistance, which causes a diode
voltage across the plasma. This diode voltage can also accelerate the charged particles along the
plasma column [11]. In a plasma focus, an anomalous resistance is one of the main mechanisms
for plasma heating leading to neutron production [12].
The typical discharge current and tube voltage (measured at the base of the DPF) signals
in a plasma focus can be seen in Fig. 1.5. The change in plasma impedance during the compression
phase can cause a sharp drop in the discharge current. The duration of the current drop also shows
the lifetime of the plasma column in the compression phase.
2
1
3
Anode Cathode
Insulator
B
8
Fig. 1.5: Typical discharge current and tube voltage traces during focusing of plasma. The
current drop and voltage peak indicate the occurrences of a focusing event.
The hemispherical plasma focus, Mather-type plasma focus, and Filippov-type plasma
focus are the three types of DPF devices. The ratio of the anode radius (a) to the anode length (L)
are very different in Mather type and the Filippov type. In the Mather type, the ratio 𝑎
𝐿 is less than
1 while in the Filippov, this ratio is greater than one. Fig. 1.6 shows the Fillipov-type of the DPF
device. In a hemispherical plasma focus the anode and cathode are two hemispheres that are
insulated electrically from each other.
In a Filippov-type, the plasma focus break-down and compression phases are the two main
phases. In a Mather-type, the plasma break-down, axial, and compression phases are the three main
phases. In a hemispherical plasma focus, the break-down phase and the combined axial
compression phases are the main two phases. In all types of the DPF device, a plasma pinch has
t(µs)
Typical voltage and discharge current signals in a DPF
t(µs)
9
to happen when the discharging current reaches to its maximum to fully use the energy stored in
the capacitor bank.
Fig. 1.6 schematics of Filippov-type dense plasma focus.
1.3 Research History on Dense Plasma Focus
The dense plasma focus has been studied for more than half a century, which has led to
thousands of scientific reports. These studies on a DPF device can be divided into six aspects:
Plasma dynamics, particle emission (electrons and ions), x-ray radiation, neutron radiation, and
fusion energy. In the following Sections, a brief review of those studies will be presented.
1.3.1 Plasma Dynamics
The formation and acceleration of the plasma in various stages of the plasma focus are
important phenomena occurring in a DPF device. It has been shown that the velocity of the current
sheath plays a key role in the neutron emission in DPF devices [4]. Formation of the plasma in the
break-down phase can also affect the focusing of the plasma in the compression phase [13]. In the
plasma focus, the current sheath moves roughly like a rigid body. This means that the dynamics
of one part of the layer depends on the other parts within the layer. A magnetic probe can be used
to study the formation and acceleration of the current sheath in a plasma focus.
Anode
Cathode
10
One of the well-known models to simulate the dynamics of plasma is the Lee model [14-
15]. By coupling thermodynamics and electrodynamics of the plasma in a plasma focus, the Lee
model has been used extensively for different plasma focus devices with different stored energies
[12, 13, 16-19]. In this model, three phases have been considered including the plasma dynamics
in the compression phase of plasma focus. The detail of the Lee model will be explained in the
next chapters.
The effect of the insulator length, types of operating gas, filling pressure, charging voltage
and the geometry of the electrodes on the dynamics of the plasma in the break-down phase, axial
phase, and compression phase have been studied experimentally in low energy plasma focus
devices [13,20-22].
In the break-down phase, the required time for the formation of the plasma layer across the
insulator depends on the charging voltage, operating pressure, and insulator length [20, 21].
Decreasing the pressure and increasing the insulator length make the formation time of the plasma
layer longer. Under a good focusing condition (i.e. significant current drop during the pinch phase),
the formation time of the plasma layer in the breakdown phase is about 100 to 300 ns. The radial
velocity of the plasma layer in the break-down phase depends on the filling gas pressure. Under a
good focusing condition, the measured radial velocity of the plasma layer is about (1-3)×104 m
s .
The focusing condition of the plasma also depends on the insulator length.
In the axial phase the anode radius, operating pressure and charging voltage, can change
the axial velocity of the plasma layer [13, 21]. Decreasing the anode radius and operating pressure,
and increasing the charging voltage, enhance the axial plasma layer velocity. In a good focusing
11
condition, the axial velocity of plasma is about (1-2) ×105 m
s. In the radial phase the measured
plasma layer velocity reaches up to 3×105 m
s .
1.3.2 Charged Particle Emission
Due to the different applications of energetic charged particle beams, such as lithography
[5], short-lived radioisotopes production [9], and thin films deposition [8], a series of experiments
have been carried out to measure and optimize the energy spectrum and the number of the charged
particles per shot, generated by DPF devices. Fig. 1.7 shows the directions of the charged particle
beam emission from a DPF device. Ions and electrons in a plasma focus are accelerated in the
opposite directions. The accelerated ions and electrons have different energy spectrums. The
energy of the generated ions is about 10 keV to 10 MeV. The power density of the generated ion
beam by the pinching plasma is about 1012 W
cm2 . The energy of the electrons is about 10 keV to
1 MeV. The power density of the generated electron beam reaches1013 W
cm2 . The electron and ion
current is on the order of discharge current during the pinch phase. The total energy of the
generated charged particle beam can be expressed based on the fraction of the total energy stored
in the inductance of pinching plasma (i.e. 0.5 L I2, where L is plasma inductance and I the plasma
current during the pinch phase) .
12
Fig. 1.7: The directions of charged particle beams emission from DPF device.
In a plasma focus, ions and electrons are accelerated by an electric field to high energies
when they become runaway (i.e. drag force acts on particle is not significant ). The generation of
run-away charged particles in a plasma can be explained by the Dreicer Theory [23]. Based on
theory, an electron becomes runaway if the component of the electric field along the magnetic field
exceeds the collisional friction force. For an electron in an electric field the equation of motion can
be presented by Eq. 1.1
𝑚𝑒𝑑𝑣
𝑑𝑡= 𝑒𝐸 − 𝐹(𝑣) (1-1)
where 𝑚𝑒 is the mass electron, 𝑣 is the electron velocity, E is the component of electric field
along the magnetic field, and 𝐹(𝑣) is the friction force that is experienced by the electron. The
friction force can be presented by Eq. 1-2
𝐹(𝑣) = 𝑚𝑒 𝒗 𝜈𝑒𝑒 (1-2)
Electron
Ion
13
where 𝜈𝑒𝑖 is the collision frequency of the electrons. By considering 𝜈𝑒𝑖 ≈ 1
𝑣3 the friction
force 𝐹(𝑣) ≈1
𝑣2. Eq. 1-1 shows that if there are no other loss mechanisms, and 𝑚𝑒 𝑣 𝜈𝑒𝑒
𝑒≤ 𝐸 , the
electron velocity is continuously increasing, and consequently the friction force that is experienced
by electron becomes negligibly smaller and smaller. Hence, runaway electrons are generated if
𝐸𝐶 ≤ 𝐸, where𝐸𝐶 = 𝑚𝑒 𝑣 𝜈𝑒𝑒
𝑒. Due to the higher mobility of electrons compared to ions, an
electron becomes runaway in a much shorter time compared to ions.
In the following chapters, the generation of runaway charged particles in a plasma focus
will be explained and discussed.
In a plasma focus, three main mechanisms have been proposed to explain the generated
electric field across the plasma column. The most common mechanism responsible for accelerating
the ions and electrons is the inductive electric field [24-26] due to the rapid local changes in the
magnetic flux accompanying the onset of the radial phase in the plasma focus device. The induced
voltage across the plasma column is expressed by 𝑉 =𝑑(𝐿𝐼)
𝑑𝑡 , where L is plasma circuit inductance
and I is the plasma current. The induced electric filed mechanism explains the generation of
energetic ions and electrons at the onset of radial phase.
The occurrence of the axial symmetrical (or 𝑚 = 0 mode) instability also induces a
plasma voltage locally across the plasma column that can accelerate ions and electrons at the end
of the compression phase.
Another proposed mechanism is the anomalous resistivity effect [12, 27-28]. In the
pinching plasma, resistance is increasing while the plasma current remains almost constant.
Therefore, the voltage across the plasma column increases. This mechanism explains several bursts
14
of energetic electrons and ions observed during the pinching of the plasma and at the post pinch
phase [12].
A mechanism which can explain the energy of electron up to the order of MeV is the
rupture of the conductivity current due to 𝑚 = 0 instability and the appearance of the current
displacement [29]. After losing plasma conductivity, the energy in the magnetic field is converted
to electromagnetic waves. The voltage across the plasma column generated by breakup of the
plasma column reaches up to 60 MV. Breakup of pinch induces a huge voltage across plasma that
can be counted for by including the displacement current ( 𝜕𝐸
𝜕𝑡 ). This mechanism also explains the
generation of hard x-rays during the pinching phase.
Magnetic reconnecting during the pinching of the plasma column might also be a possible
mechanism to induce a huge voltage across the plasma column to a value about 10 MV. The
magnetic reconnection occurs within a time interval of 1 ps in a narrow gap of about (1-10 µm)
[30].
The main detectors that measure the generation of electron and ion beams from a DPF
device includes Rogowski coil [31], a Faraday cup [12, 32], a magnetic electron energy analyzer
[33], and a Cherenkov detector [34].
1.3.3 X-ray Radiation
There are two main, and well-known, mechanisms for x-ray radiation in plasma focus
devices: line and continuum radiation. The line radiation is generated by a working gas, or from
the interaction between the energetic electrons and impurities. The line radiation generated from
the impurities in a pinching plasma has a significant contribution in the total radiation.
15
In DPF machines, continuum includes recombination and Bremsstrahlung radiation. The
latter is caused by the accelerating electrons in the Coulomb field of ions, producing soft and hard
x-rays. The interaction of the electron beam with anode tip also generates hard x-rays with energies
up to 1 MeV.
The soft x-ray energy in a plasma focus is in the range of 1 keV to 10 keV. The duration
of the soft x-ray radiation in a plasma focus is approximately hundreds of nanoseconds. The
generated x-ray power is about 108 W
cm2 at the pinch while the duration of hard x-ray generation
from a pinching plasma is about tens of nanoseconds. The energy of hard x-ray is in the range of
100 keV to 1 MeV.
In a plasma focus, the m = 0 instability and the anomalous resistance are also the
mechanisms of hard x-ray generation [12]. The anomalous resistance mechanism also explains the
multiple intensity peaks of hard x-rays from a pinching plasma [12].
A plastic scintillator along with a photo multiplayer tube (PMT) and Si-based diodes, have
been used to measure the energy and spectrum of the radiated x-ray from the pinching plasma.
The energy spectrum of the radiated soft x-ray also gives valuable information on plasma
temperature [35, 36].
1.3.4 Neutron Emission
The production of neutrons in a plasma focus is based on two main mechanisms: the
thermal mechanism and the beam target mechanism [37]. The thermal mechanism of neutron
production is based on the collision of energetic deuterium ions inside the bulk of the plasma. The
16
beam target mechanism in a plasma focus is caused by the interaction of accelerated deuterons
with the plasma or background gas.
In a plasma focus, the beam target generates the most significant part of neutrons. The
yield of neutron generation based on this mechanism (Yb-t) can be expressed by Eq. 1-3 [16, 18].
(1-3)
where ni is the ion density, b is the cathode radius, rp is the radius of the plasma pinch with length
zp, σ is the cross-section of the fusion reaction, and U is the beam energy in SI units. Cn is a
calibration constant and is in the order of 107 [17].
In a DPF device, increasing the stored energy of the plasma focus can enhance the number
of charged particles and the number of neutrons per shot. The neutron yield per shot in a low
energy plasma focus is proportional to E2. In the high stored energy plasma focus, the neutron
yield is proportional E0.8. The neutron yield will reach a saturation level by increasing the energy
of the plasma focus to a higher stored energy [18].
The typical number of neutrons that can be produced by a 1 kJ plasma focus is about
108 (neutron
shot). It should be noted that increasing the energy of the plasma focus device can be done
by increasing the energy of the bank through increasing the charging voltage and capacitance of
the bank. There is an optimum limit for a breakdown voltage across the insulator. Increasing the
charging voltage over a certain experimental limit (50 kV) can cause a break-down in all space
between the electrodes, and prevent a proper formation of the current layer across the insulator.
Increasing the capacitance may also cause the discharge time to become longer, which allows
growth of instability through the plasma current before the compression phase. The growth of
Y b-t= Cn ni(Ipinch)2(zp)
2 ln (𝑏
𝑟𝑝) σ U -0.5
17
instability prevents proper formation of the plasma column in the final stage of the plasma focus.
Due to the above limitations of increasing the stored energy for the plasma focus, increasing the
neutron yield by increasing the stored energy of a plasma focus is not practical.
1.4 Dense Plasma Focus Applications
Dense plasma as a low cost and intense source of charged particles, x-rays, and neutron
has been used for different industrial and medical applications. In the following subsections three
main applications of DPF will be presented and discussed.
1.4.1 Short-lived Radioisotopes Production
The current method to produce short-lived radioisotopes (SLRs) for medical applications
is through a cyclotron. The operation costs of a cyclotron can go up to millions of dollars.
Furthermore, neutron emission from a cyclotron raises up safety environmental hazards.
Therefore, an alternative technology is desired to produce SLRs in a safe and cost-effective way.
Production of SLRs, such as 13N, 17F, 18F, 15O, and 11C, through a plasma focus is one of
the promising medical applications in the medical sciences [38-44]. SLRs can be produced in a
plasma focus through the bombardment of an external solid (exogenous method) or a high atomic
number gas (endogenous method) targets by energetic ions as can be seen in Fig. 1.8.
18
Fig. 1.8: The bombardment of target by ion beam generated by plasma.
These short-lived radioisotopes are positron emitters used for positron emission
tomography (PET) imaging. It should be noted that due to its short lifetimes in these isotopes, a
high number of energetic ions is needed to activate the target to a suitable level for medical
applications within a time less than the half-life time of the isotopes. The main problem in using
dense plasma focus devices for SLR production is that only a limited number of energetic ions can
contribute to isotope activation. For instance, the lifetime of N-13 as one the important SLRs used
for medical applications is about 10min, and the level required for activation is 4 giga Becquerels
(GBq). Such level of activation is far greater than the maximum expected activity that one could
achieve in a single shot of a medium energy plasma focus. To solve this issue, the repetitive
operation of a plasma focus with a high stored energy is needed. However, technical issues, such
as heating and high level of impurities, may arise during the repetitive operation of a high energy
DPF device.
Ion Target
19
The highest reported activity of N-13 produced in a carbon target is about 200kBq, and
was achieved by using a 76 kJ DPF device operating in a single discharge mode [44]. In general,
the activation rate of nitrogen-13 in a DPF can be expressed by the Eq. 1-4 [41]:
𝑁13 = 𝐾 ∫ 𝑌𝑡𝑡 𝐸−𝑛 (𝐸)𝑑𝐸
𝐸𝑚𝑎𝑥
𝐸𝑚𝑖𝑛 (1-4)
Where N, E, Ytt are the number of activated nuclei, the energy of the ions, and a thick target
yield of production, respectively. The values of unit less n and K are 5 and 5.95 × 1011 MeV-1.
𝐸𝑚𝑖𝑛 and 𝐸𝑚𝑎𝑥 refer to the threshold energy for activation and maximum energy of ions ejected
from the plasma focus. In the above equation, 𝐾 𝐸−𝑛 represents the energy distribution function
of the ion beam produced by a DPF, where n ≃2. The energy distribution function shows that the
number of deuterium ion decrease by increasing the ion energy.
Based on the maximum activity in low and medium energy machines, such as NX-2
(1.7 kJ) and PF-150 (operated in 20 kJ) devices, the number of high energy ions in the order of
(1 − 2) × 1012 and (1 − 2) × 1013 have been reported respectively [41, 43]. These numbers have
been interpreted from the measured nitrogen-13 activity. The activity of the nitrogen-13 nuclei, in
Bq, can be estimated by the Eq. 1-5 where 𝑁𝑜𝑑 is the number of high -energy deuterons with energy
over 340 keV[41].
𝐴 = 10−9𝑁𝑜𝑑 (1- 5)
In a plasma focus, 1015 ions can be produced per kilojoules of stored energy in a capacitor
bank [45]. However, only a small portion of these ions (≃10-3Nion) can contribute to the activation
of the 12C. The minimum required energy for the N-13 activation through the 12C (d, n)13N reaction
is about 340 keV, and the cross-section reaches its maximum at 2.5 MeV. By considering Eq. 1-4,
20
1 GBq activation of N-13 can be achieved by a 1 GJ DPF device in a single shot. As it is previously
mentioned, operating a DPF device with such a high energy is not practical.
Repetitive operation of a DPF device might decrease the required stored energy of device.
In the case of N-13 with a half lifetime of 10 min, the operation of a device with a stored energy
in the order of 102 kJ, and with the repetitive rate on the order of 10 Hz, is needed to produce
1 GBq of N-13. It should be noted that operating a repetitive DPF reduces the neutron yield per
shot in a plasma focus. Therefore, a high repetition rate may not be the only solution for SLR
production [46].
By considering the above technical issues, shifting the energy spectrum and numbers of
ions to higher values would be an ultimate solution for SLR production in a plasma focus. It should
be noted that the number of ions produced by a 1 kJ plasma can produce 1 MBq in a single shot if
all produced ions can contribute to the activation process. Based on these issues, this thesis will
study the effective parameters on ion beam emission from a plasma focus.
1.4.2 Thin Film Deposition and Ion Implantation
A dense plasma focus as a source of charged particles can be used for thin film deposition
[47-49], ion implantation [50-52], and phase change of thin films [53-54]. Deposition of different
materials on a substrate can be done in a plasma focus by using the electron beam and the ion beam
generated from the pinching plasma. In the case of electron beam, a target is placed on the tip of
anode as seen in Fig. 1.9. The generated energetic electron beam hits the anode tip and sputters the
target. The sputtered material gets deposited on a substrate that is placed on the top of the anode.
In the second scenario the target is placed on the top of anode. The generated ion beam from the
21
pinching plasma hits the target and sputters it on a substrate that is placed next to target. Deposition
of TiN and synthesizing of CoPt on Si substrates have been done using a low energy plasma focus
[55-56].
Fig. 1.9: Schematic arrangement for thin film deposition in a plasma focus.
A dense plasma focus can also be used for ion implantation and nitrating of different
materials. Nitrating of Si [57], Aluminum [58], Titanium [59], and Tungsten [60] in a low energy
plasma focus has been done in many reported studies. For this purpose, a substrate is placed in
front of the hollow anode. Nitrogen has to be used as filling gas. The generated energetic nitrogen
ions from the pinching plasma penetrates into the substrate, nitrating the substrate. The graphite
carbon target on the top of the anode is also used for generating carbon ions for formation of TiC
in a plasma focus [62]. Proton beams can also be generated by using hydrogen as a filling gas in a
plasma focus. The proton beam is used for annealing ion-implanted semi-conductors [61]. Phase
Substrate
Target Plasma
22
changing of thin film is also done by using noble gases such as argon. For this purpose, a noble
gas has to be used as filling gas in a plasma focus [53].
The distance of the substrate from the pinching plasma, angle of the substrate with respect to
the axis of the pinching plasma, and the number of shots, are important parameters in thin film
deposition and implantation processes. Decreasing the distance between substrate and anode tip
would increase the thickness of the deposited thin film per shot. The thickness of the deposited
thin film also increases by increasing the numbers of shots. Increasing the angle of the substrate
with respect to the pinch axis also decreases the thickness of deposited thin film per shot in the
plasma focus. In ion implantation process, increasing the angle of substrate with respect to the
pinch axis and increasing the distance from pinching plasma would decrease the flux of the
incident ion beams per surface area.
1.4.3 Detection of Illicit Materials and Explosives
Detection of illicit and explosive materials is one of the important applications of a neutron
source [63-65]. Interaction of neutrons with different materials produces numbers of scattered
neutrons and gamma rays. The scattered neutrons and gamma rays are produced because of the
elastic and inelastic scattering of the primary neutrons with the nuclei of the irradiated matter.
The energy of the scattered neutrons in elastic interaction with different nuclei would be
different due to the different masses of the nuclei. The generated gamma rays due to inelastic
interactions of neutrons with the nuclei also give valuable information on elemental composition
of the sample. Oxygen, nitrogen, carbon, hydrogen are the main elements that exist in explosive
materials.Due to the low intensity of neutron sources (≤ 10 8 neutron
shot) and the required long time
23
(few minutes to half an hour) in detecting the elemental composition of the object [66-68], an
alternative neutron source that produces an intense pulse of neutrons is in demand.
The dense plasma focus is a cost-effective candidate of neutron source that produces up to
1012 fast neutrons within tens of nanoseconds. Such intense neutron source would reduce the
required time for detecting the elemental composition of samples. The required short time and high
accuracy of results due to the low level of noise to signal ratio also prevents the activation of
exposed materials [69-70]. A dense plasma focus with a stored energy of 7 kJ, operating in D-T
along with several neutron detectors (plastic scintillator + photomultiplier tube (PMT)) were used
for this purpose [69-70]. The schematic arrangement of the experiment can be seen in Fig. 1.10.
Fig. 1.10: Schematic arrangement of illicit and explosive materials detection by a DPF.
DPF device H,C,N,P
PMT PMT
Fusion neutrons
Neutron and x-ray Shieling
Scattered
neutrons
24
1.5 Motivation of the Thesis
Due to the important applications of charged particle sources in science and modem
technologies, developing a cost-effective ion and electron beam source has a tremendous value.
The primary goal of this thesis is to enhance the efficiency of a DPF device as an energetic charge
particle source for the prodcution of isotopes.The main work done is listed below:
A low energy Mather type plasma focus (DPF UofS-I) with a stored energy up to
3 kJ has been developed. DPF UofS-I has been optimized as an ion source by
incorporating existing experimental results and theoretical models.
By analyzing the discharge circuit of DPF UofS-I plasma resistance has been
measured.
The effects of anomalous resistance and plasma heating on run-away charged
particles generation from pinching have been studied.
The effects of atomic number of the gas on plasma inductance and resistance have
been investigated.
The effect of atomic number of the gas on ion emission from the pinching plasma
has been explored.
A new type of dense plasma focus has been developed.
The experimental results regarding the ion beam emission from different operating gases
have been compared with simulation results based on the Lee model.
25
1.6 Thesis Outline
Chapter 1 covered the introduction of the plasma focus. The physics of charged particle
radiations from the pinching plasma was discussed. Different types of plasma focus devices were
introduced, and the limitations of using a plasma focus to produce fusion energy were presented.
Short-lived radioisotope production by a plasma focus, and the technical difficulties to utilize a
plasma focus as an ion source for this purpose, were explained.
In chapter 2, the Lee model, as a theoretical model to simulate operation of DPF UofS-I as
an ion source, will be introduced.
Chapter 3 will be on the design and fabrication of the DPF UofS-I. The technical details
regarding the building process of the DPF UofS-I will be presented and discussed. The
optimization of the device as ion source based on Lee model will be explained.
Chapter 4 will be on the plasma diagnostics that have been designed and installed on the
DPF UofS-I. The technical details regarding the design and fabrication of a Faraday cup as an ion
beam detector, an electron beam extraction and measurement setup, a soft x-ray detector, and a
hard x-ray detector, will be explained.
Chapter 5 will highlight the key results on energetic charged particle emission from a dense
plasma focus device. By analyzing the discharge circuit, a new method for analyzing the
anomalous plasma resistance as the main mechanism to generate energetic ions and electrons will
be presented. By measuring the plasma voltage and inductance, the required electric field for
runaway charged particle generation from the pinching plasma will be discussed. The Driecer
theory will be considered, and the avalanche process effects of plasma heating required in electric
26
fields for runaway charged particle generation from the pinching plasma will be presented. It will
be shown that the required electric field for runaway electron generation in the pinching plasma
and Tokamaks are similar, even though the two types of devices are working on different time
scales. The results regarding the current density and energy of ions generated from the DPF UofS-
I as an optimized ion source will be presented and compared with the Lee model. To enhance the
ion beam and x-ray generation from a plasma focus device, a new type of dense plasma focus will
be introduced. Based on the electrical discharge signals, the generation of several focusing events
in the new type of plasma focus will be discussed.
In chapter 6, the major results will be summarized, and possible future work will be
presented.
27
Chapter 2
PHYSICS OF Z-PINCH
2.1 Introduction
The physics of Z -pinch is explained by considering MHD equations to predict the temporal
behavior of the plasma current, temperature, and radius. By assuming: a constant line density of
plasma column, thermal equilibrium of ions and electrons, quasi neutrality of plasma, and an
isothermal plasma column, the MHD momentum equation for a fully ionized plasma can be
expressed by Eq. 2-1.
𝑚𝑛𝐷𝒗
𝐷𝑡= −𝛁𝑃 + 𝑱 × 𝑩 − 𝛁 ∙ ᴨ (2-1)
where m is the mass, n is the particle density, V is velocity, P is the scalar diagonal pressure, J is
the current density, and ᴨ is the off-diagonal pressure tensor. All elements in Eq. 2-1 are a function
of time and radius for an axial symmetric system such as DPF.
The right-hand side of Eq. 2-1 shows the force from the plasma pressure, magnetic
pressure, and the viscous effect. The left-hand side of Eq. 2-1 shows the inertial momentum. The
derivative 𝐷
𝐷𝑇 is the convective derivative (i.e.
𝐷
𝐷𝑡 =
𝑑
𝑑𝑡+ 𝐯 ∙ 𝛁 ). By ignoring the inertial term
and the viscous term (considering plasma is homogenous and is in steady state), Eq. 2-1 gives
𝛁𝑃 = 𝑱 × 𝑩 (2-2)
28
where J is axial current vector and B is azimuthal magnetic field vector in cylindrical coordinates.
Maxwell’s equation with no displacement current in cylindrical coordinates gives
𝛁 × 𝑩 = µ 𝑱 = 1
𝑟
𝜕
𝜕𝑟 (𝑟 B) 𝒆𝒛 (2-3)
where µ is the magnetic permittivity of plasma and ez is the unit vector in the direction of plasma
column or the axial direction. The cross product of Eq. 2-3 by 𝐵𝒆𝜃 gives
𝑱 × 𝑩 = − 𝐵
µ𝑟
𝜕
𝜕𝑟 (𝑟𝐵) 𝒆𝒓 = −
1
2µ𝑟2
𝜕
𝜕𝑟 (𝑟2 𝐵2 ) 𝒆𝒓 (2-4)
The equation of state gives
2e e i iP n K T n K T nKT (2-5)
where K is the Boltzmann’s constant, T is the temperature, and the subscripts i and e refer to
electrons and ions. It is also assumed that e in n n and e iT T T . The left-hand side of
Eq. 2-2 in cylindrical coordinates can be expressed by Eq. 2-6.
𝛁𝑃 =𝜕
𝜕𝑟 (𝑃 ) 𝒆𝒓= 2 KT
𝜕
𝜕𝑟(𝑛) 𝒆𝑟 (2-6)
Substituting of Eq. 2.4 and Eq. 2.6 in to Eq. 2-2 gives
2 𝐾𝑇𝜕
𝜕𝑟(𝑛) 𝒆𝒓 = −
1
2µ𝑟2 𝜕
𝜕𝑟 (𝑟2 𝐵2 ) 𝒆𝒓 (2.7)
By multiplying both sides of Eq. 2-7 by 𝜋𝑟2 and integrating over the radius, the left side of
Eq. 2-7 gives where N is the linear density with a unit of number of particles
m.
∫ 2𝜋𝑟2 𝐾𝑇𝜕
𝜕𝑟(𝑛) 𝑑𝑟
∞
0 = −2𝐾𝑇 ∫ 2𝑛𝜋𝑟𝑑𝑟
∞
0 = −2𝐾𝑇𝑁 (2-8)
29
It should be noted that in the above integration, it is assumed that n (r = ∞) = 0.
Integrating the right-hand side of Eq. 2-7 gives
− ∫1
2µ𝜋
𝜕
𝜕𝑟 (𝑟2 𝐵2 ) 𝑑𝑟
∞
0=
µ2𝐼2
8𝜋2 (2-9)
By considering Eq. 2-8 and Eq. 2-9 the momentum equation gives Eq. 2-10, which is called the
Bennet equilibrium equation. In Eq. 2-10, I is the plasma current. This equation expresses the
balance between magnetic and thermal pressures and can be used to estimate the plasma
temperature.
µ2𝐼2
8𝜋2 = 2𝐾𝑇𝑁 (2-10)
The thermal pressure of plasma column can change due to thermal heating and radiation loss.
Temporal variation of plasma thermal pressure can be explained by MHD energy equilibrium.
3
2
𝐷𝑃
𝐷𝑡+
5
2𝑃𝛁 ∙ 𝑽 = ɳ 𝐽
2 − 𝜵 ∙ 𝒒 − 𝑅 (2-11)
where ɳ is the classic plasma resistivity, q is the heat flux, and R is radiation loss. The first
term in left hand side of Eq. 2-11 is the change in internal energy of the plasma and the second
term is the work done by plasma during expansion against magnetic pressure. The first term in the
right-hand side of Eq. 2-11 is Joule heating, and second term in right hand side is heat flux. At a
steady state equilibrium Eq. 2-11 reduces to Eq. 2-12
𝑅 = ɳ 𝐽2 (2-12)
Radiation loss is mainly due to Bremsstrahlung and line radiation in the pinching plasma. The
Bremsstrahlung radiation is more important in hydrogen-like atoms and depends on plasma density
and temperature, R≈ 𝑛2 𝑇1
2 . Classical resistance of plasma depends on plasma temperature,
30
ɳ ≈ 𝑇−3
2 . Eq. 1-12 also defines a critical current (peace current) that balance the Joule heating
and Bremsstrahlung radiation loss.
In a Z-pinch, due to intense radiation of x-ray and the occurrence of anomalous resistance
due to micro instabilities and turbulences, the Joule heating and Bremsstrahlung radiation loss
might not stay balanced. The unbalanced energy gain/loss process in these spots produces macro
instabilities characterized by the perturbations expressed in the temporal and angular components
form (in cylindrical coordinate system) 0 sinm m m
m
x x x m t , where x represents
any physical quantities, such as density, pressure, or magnetic field, 0x the equilibrium
distribution, m the poloidal mode number, and mx , m , and m the perturbation amplitude,
frequency and the constant phase associated with the mode m . The macro instabilities such as
m=0 (symmetric sausage mode) and m=1 (kink mode) can be seen in Fig. 2.1. The magnetic
pressure is expressed by 𝐵𝟐
𝟐µ =
µ2𝐼2
8𝜋2𝑎2 where B is the magnetic field caused by the plasma current.
Fig. 2.1: Macro instabilities such as m=0 and m=1 through the plasma column.
𝑩𝟐
𝟐µ
In
crea
sin
g
𝑩𝟐
𝟐µ
Dec
rea
sin
g
Cu
rre
nt
Cu
rre
nt
𝑩𝟐
𝟐µ
In
crea
sin
g
𝑩𝟐
𝟐µ
Dec
rea
sin
g
31
The resistive diffusion of magnetic field into plasma is another reason that terminate
plasma confinement when plasma resistance becomes significant. The temporal and spatial
evolution of magnetic field can be expressed by Eq. 2-13, where v is ion velocity, ƞ is plasma
resistance, and µ0 is vacuum magnetic permittivity.
𝜕𝑩
𝜕𝑡= 𝛁 × (𝒗 × 𝑩) +
ƞ
µ0∇2𝐵 (2-13)
The first term in the right-hand side of Eq. 2-13 shows convection term. The second term
in the right-hand side of Eq. 2-13 is resistive diffusion term. In the case of hot and dense plasma
when the plasma resistance is negligible the magnetic field is frozen into plasma and diffusion of
magnetic field in to plasma occurs in a time much longer than the confinement time.
The electrodynamic of plasma in z-pinch machine can be explained by some model such
as snowplough and slug models. In snowplough model the current flows in a skin layer which acts
as a piston. As the piston moves it produces a supersonic wave in front of plasma skin layer. A
fraction of the ionized gas is accumulated by the supersonic wave in this skin layer as in a
snowplough [1]. In the slug model the movement of current layer produce a supersonic wave that
ionizes the gas in front of the piston. In slug model, it is assumed plasma reaches to a minimum
radius when the supersonics wave reaches to the central axis [1]. One well-known model that is
based on snowplough model and slug the model is Lee model [14, 15]. In the following part the
Lee model will be explained.
32
2.2 Lee model
The Lee model is a well-known model that has been used in different Mather-type and
Fillipov-type plasma focus devices to predict the dynamics of the plasma, production of neutrons,
and emission of charged particles from the pinching plasma [14, 15]. The model incorporates the
plasma electrodynamic and thermodynamics equations, and the geometry of electrodes, trying to
reproduce the experimental electrical signals (i.e. discharge current and voltage between anode
and cathode) and predict the output of plasma focus as a neutron and charged particle source. In
the Lee model, the break-down, axial, and compression are the three main phases of plasma
dynamics in a Mather-type plasma focus device. The compression phase is divided into: inward
radial shockwave phase, outward reflected shock wave phase, and slow compression phase. In the
following sections, a description of each of the phases will be presented and explained.
2.2.1 Axial Phase
In the Lee model, the axial phase is initiated after the break-down phase. In this phase, the
current layer that is formed in the break-down phase is accelerated by the J × B force in the axial
direction, as can be seen in Fig. 2.2. An umbrella shaped current makes a shock wave (SW) in its
front due to the magnetic pressure (MP). The shockwave makes the gas ionized and forms a plasma
layer on its backside as can be seen in Fig. 2.2.
33
Fig. 2.2: The formation of plasma layer in plasma focus.
The plasma layer carries the fraction of the mass of the gas (fma) that is filling the space
between the anode and cathode from the bottom to the top of the discharge electrodes. Z is the axial
position of the current layer from the base plate. By assuming Z=0 as an initial position of the
current layer, Z=Z 0 as the final position of current layer, n0 as number of atoms per volume, m as
the atomic mass of the filling gas, a as the anode radius, b as the cathode radius, and f ma as a
fraction of mass that carry by plasma, the total mass of plasma can be estimated by
22 abmznf ma . Fig. 2.3 shows the axial phase in Mather-type plasma focus.
Fig. 2.3: The axial phase in a Mather type plasma focus.
Z
B
MP
SW Anode
Plasma
Cathode Gas
2b
B Anode
Cathode 2a
Z=0
Z=Z0
34
The voltage across the discharge tube (Vtube) can be expressed by Eq. 2-14,
𝑉𝑇𝑢𝑏𝑒 =𝑑 (𝐿(𝑡)𝑓𝑐 𝐼)
𝑑𝑡+ 𝑟𝑓𝑐 𝐼 = 𝑉0 −
∫ 𝐼𝑑𝑡𝑡
0
𝐶0 − 𝐿0
𝑑𝐼
𝑑𝑡− 𝑅0𝐼 (2-14)
where fc is the fraction of discharge current that pass through plasma, V0 is the charging voltage
of the capacitor, C0, L0, R0 are the capacitance, static inductance, and static resistance of the
discharge circuit. I is the discharge current, while L(t) and r(t) are time dependent inductance and
resistance of the discharge tube, respectively.
By assuming that the inductance of tube L(t)= μ/2π z (t) ln (b/a) in the axial phase, Eq.
2-14 gives the time derivative of the discharge current where c= b/a.
0 0 0
0
( ( )) ln ln2 2
cc
dI fIdt dzV R r t I If c L c z
c dtdt
(2-15)
By considering the plasma as a rigid body the total magnetic force that pushes the plasma
in the axial direction can be expressed by Eq. 2-16
2
ln2 2
b pPz p
a
II bF I dr
r a
(2-16)
where IP = fc I and fc is an input fitting parameters in the model.
By considering the mass of the plasma, the momentum equation in the axial phase can be
expressed by Eq. 2-17,
2 2 2 2
0 0
( ) 1z ma ma
d mV d dz d dzF b a z f c a f z
dt dt dt dt dt
(2-17)
35
where 0 0n m is the mass density, c= b/a , and Z is the distance of plasma sheath from the
baseplate. By substituting Eq. 2-16 in Eq. 2-17, the momentum equation can be expressed by Eq.
2-18 where
2
2 2
0 1 ln2
p
ma
Id dzc a f z c
dt dt
(2-18)
Assuming that the plasma current arrives at the top of the anode at the time, ta, from the
start of the discharge. This characteristics time depends on the anode length Z0 among other
parameters and can be derived from Eq. 2-18 by considering 𝑑𝑧
𝑑𝑡=
𝑧0
𝑡𝑎 where ta is the duration of
axial phase when the current reaches to the top of the anode, z = Z0, and IP =fc I.
12 2 2
0
0
4 ( 1)
ln ( )
ma
a
c
f Zct
Ic fa
(2-19)
The choice of the characteristics time ta should match the discharge time of the capacitor
bank ( √𝐶0𝐿0 ). Eq. 2-19 gives an estimation of a proper electrode geometry and operating pressure
of filling gas, based on the discharge time of the capacitor bank. It should be noted that Eq. 2-15
and Eq. 2.18 are coupled equations. The change in plasma current ( 𝑑𝐼
𝑑𝑡 ) depends on
d𝐿(𝑡)
d𝑡 and the
change in the tube inductance depends on the plasma current (i.e. d𝐿(𝑡)
d𝑡α
𝑑𝑍
𝑑𝑡 α 𝐼𝑃
2). Lee solved the
two coupled equations 2-15 and 2-18 numerically for z and I to reproduce the discharge current
waveform in the axial phase using the known electrical constants C0, V0, R0, 𝜌0 (based on the
36
operating pressure) and the geometry of electrodes, and also assuming vanishing resistance r(t)≃0
during the axial phase as a very good approximation. Afterwards, the voltage waveform V(t) can
be solved by combining Eq. 2-16 and Eq. 2-17.
2.2.2 Inward Shockwave Phase
When the plasma layer arrives at the top of the anode, the J × B force pushes the plasma
layer into the center of the anode. The first stage in the compression phase is the inward radial
shock phase. In this phase, a plasma column with the inner radius of rs and the outer radius of rp
(Fig. 2.4) will form on the top of the anode. The rs shows the shockwave distance from the center
of the anode and rp shows the outer radius of plasma column from the center.
Fig. 2.4: The inward radial shock wave in final stage of plasma focus.
In this phase, the magnetic compression pressure caused by the J × B force on the outer
side of plasma column creates a shockwave inside the plasma column. The shockwave moves into
the center of the anode and compresses the gas inside the plasma column. The shockwave will
leave the ionized gas in its backside that forms a plasma layer in the radial phase. The plasma layer
Anod
B 2r p
2rs
37
moves radially into the center of plasma by a magnetic pressure. Due to the open end of plasma
column, it is expected that the shockwave also moves in the axial direction and causes an
elongation of the plasma column in the axial direction. By using Eq. 2-14, and the plasma column
inductance μ / 2π Z f (t) ln (b / r p (t)), where Zf is the length of plasma column, the time derivative
of the discharge current in the compression phase can be expressed by Eq. 2-20.
0 0
0 0
ln2 2
ln ln2 2
f f p
c c
p p
c c f
p
dZ Z drIdt μ b μV (R r(t))I f I f I
C π r dt π r dtdI
dt μ μ bL f c Z f Z
π π r
(2-20)
Eq. 2-20 shows that increasing the plasma inductance or resistance will result in a drop in
the discharge current during the compression phase.
The dynamics of the shockwave and the plasma can be explained by the theory of
shockwaves and thermodynamics of plasmas during the compression phase. As mentioned, the
magnetic pressure drives the shockwave in the filling gas. The shockwave velocity can be derived
by Eq. 2-21, where PB is magnetic pressure and Vs is the shock wave velocity (i.e. 𝑑
𝑑𝑡𝑟𝑠).
2
0
2
1B sP V
(2-21)
Where 𝛾 =𝑐𝑝
𝑐𝑣 is the ratio of the specific heats at constant pressure (cp) and volume (cv).
By substituting the magnetic pressure,
2
22
cB
p
IfP
r
, the shock velocity can be expressed
by Eq. 2-22 where f mr is radial mass fraction.
38
12( 1)
8
s c
mr p
dr f I
dt f r
(2-22)
The negative sign shows the inward movement of shockwave in the compression phase.
By assuming the time duration when the plasma current reaches to the center of anode (tr)
as the characteristics time and the anode radius (a) as characteristic length (i.e.𝑑𝑟𝑠
𝑑𝑡≈
𝑎
𝑡𝑟), Eq. 2-22
gives
aI
fmr
f
t
aV c
r
r
/
4
)1(2/1
(2-23)
Eq. 2-24 can be used to estimate the radial compression phase in a plasma focus based on
the anode radius, the operating pressure, and the discharge current.
1/2
0
4
( 1)
mr
r
c
f at
f I
a
(2-24)
The plasma column is an open-end column, and the compression of the gas in the radial
direction creates a shockwave in the axial direction. Based on the propagation theory of
shockwaves in the radial and axial directions, the axial velocity of shockwave can be expressed in
terms of radial velocity of the shockwave by Eq. 2-25.
dt
dr
dt
Zd ss
1
2
(2-25)
39
By assuming Z s ≃Z f , Eq. 2-25 can define the elongation of the plasma column in the
compression phase.
As can be seen in Eq. 2-20, the change in discharge current (𝑑𝐼
𝑑𝑡 ) column depends on the
change of the plasma radius (𝑑𝑟𝑃
𝑑𝑡). By assuming that the plasma resistance is a negligible term, the
compression of the plasma can be an adiabatic process (i.e. PVƔ = constant).
The differential from the magnetic pressure gives 𝑑𝑝
𝑝 can be expressed by Eq. 2-26
p
p
r
dr
I
dI
p
dp2 (2-26)
The differential of PVƔ = constant can be expressed by Eq. 2-27
(2-27)
Expressing the plasma volume as fsp ZrrV 22 the dV can be expressed by
Eq. 2-28.
𝑑𝑉 = 2𝜋( 𝑟𝑝 𝑑𝑟𝑝 − 𝑟𝑠 𝑑𝑟𝑠 )𝑍𝑓 + 𝜋( 𝑟𝑃2 − 𝑟𝑆
2)𝑑𝑍𝑓 (2-28)
Substituting Eq. 2-27 and Eq. 2-28 in Eq. 2-26, gives
0P
dP
V
dV
40
p
p
spf
fspfsspp
r
dr
I
dI
rrz
dzrrzdrrdrr2
2)(
1
22
22
22
= 0 (2-29)
The time rate of the change of plasma column radius can be expressed by Eq. 2-30
dt
dr
dt
dI
I
r
r
rrz
dt
dzrrz
dt
drr
dt
drr
pp
p
s
pf
f
spf
s
s
p
p
)1(
21
2
2
2
22
(2-30)
Eq. 2-20 and Eq. 2-30 are coupled equations that Lee solved numerically for I and rp by
considering electrical constants C0, V0, R0, 𝜌0 (based on operating pressure), geometry of electrodes
and Eq. 2-20 and Eq. 2-25 .
2.2.3 Reflected Shockwave Phase
The shockwave will be reflected radially in the outward direction after hitting the center
of the anode; this time would be the onset of the radially reflected shockwave phase as can be seen
in Fig. 2.5. Based on the shockwave theory, the speed of reflected shockwave can be presented by
Eq. 2-31
axison
sRs
dt
dr
dt
dr
3.0 (2-31)
The axial elongation of the plasma column will continue with the velocity expressed by
Eq. 2-32
41
axison
sf
dt
dr
dt
dZ
1
2
(2-32)
In this phase, the time derivative of the discharge current can be expressed by Eq. 2-20.
The time rate of the plasma column radius can be calculated by Eq. 2-30 by substituting 𝑑𝑟𝑠
𝑑𝑡 = 0
(as the volume of plasma column will not decrease due to reflection of shockwave), and rs ≃ 0 at
the time of the maximum compression of gas at the center of the anode by the shockwave.
Fig. 2.5: The reflected shockwave phase in plasma focus.
1
dt
dZ
Z
r
dt
dI
I
r
dt
dr
f
f
pp
p (2-33)
This phase will end when the reflected shockwave hits the plasma layer (i.e. rs = rp).
2 rs ≈ 0
2 rp
Anode
B
42
2.2.4 Slow Compression Phase
The slow compression phase starts when rs = rp as the reflected shockwave produces a
pressure in the opposite direction of the magnetic pressure. At this time, the plasma column is
compressed to its minimum radius when the inner plasma pressure reaches its magnetic pressure.
By considering the plasma density and the plasma current, and using the Bennet equation, the peak
of the plasma temperature is estimated.
The plasma resistance has to be taken into account due a small plasma column radius.
Plasma radiation is significant due to a high density and temperature of the plasma, and the
compression of plasma is not adiabatic in this phase.
Eq. 2-20 presents the time derivative of the discharge current. To find the time rate of the
plasma radius, Lee used the first thermodynamics law VdpdQdh , where h is the enthalpy of
the plasma. Considering the energy gain and loss terms, dQ can be expressed by Eq. 2-34.
jouleheating radation ion
dQ d Q Q E (2-34)
where Qjoule heating is caused by plasma resistance. Qradiation includes all the x-ray radiation for a
compressed plasma, and Eion is the amount energy that is consumed to ionize the atoms. Enthalpy
can be also expressed by Eq. 2-35
PVh1
(2-35)
By differentiating the enthalpy, the first law of thermodynamics gives
43
VdpdQPdVdPVdh
)(1
(2-36)
dQVdPpdV 1 (2-37)
By doing the same process for finding dV and dP in the first stage of the compression
phase, the time rate of the plasma column radius can be expressed by Eq. 2-38
1)1(4
1
122
dt
dQ
If
r
Zdt
dZ
Z
r
dt
dI
I
r
dt
dr
c
p
f
f
f
ppp (2-38)
The change of plasma column length in this phase will be derived by considering the internal
pressure of plasma. This pressure can create a shockwave in the axial direction with the velocity
expressed by Eq. 2-39. The maximum change in plasma column length can be expressed by Eq.
2-39 when the internal pressure of plasma reaches to magnetic pressure.
1/2
2
04 ( 1)
f c
p
dZ I f
dt r
(2-39)
Eq. 2-38 and Eq. 2-20 can be solved numerically by calculating the plasma resistance and energy
loss by x-ray radiation in this phase. Lee incorporated the classical term of plasma resistance,
power loss by the line and Bremsstrahlung radiations into the model, and predicted the dynamics
of plasma in this phase.
44
It should be mentioned that incorporating the x-ray radiation and self-absorption of plasma
Lee could predict the radiation-enhanced compression of plasma that is the subject of many
theories related to DPF [71]. Radiation-enhanced-compression is caused by a drop of internal
plasma pressure due to the extreme x-ray radiation from the plasma. In this thesis, the occurrence
of radiation-enhanced-compression will be shown experimentally.
2.2.5 Instability Phase
When the plasma reaches to its maximum compression, the plasma column may also be
unstable in a very short period of time due to the occurrence of instabilities in the plasma. The
occurrence of instabilities in plasma makes the plasma resistance anomalous in the post pinch
phase.
Lee also incorporated the anomalous resistance in the code to explain the energy
consumption after the pinch phase [45]. In this thesis, the predicted anomalous resistance in the
post pinch phase that is the subject of many types of research on plasma focus devices will be
measured experimentally. The time variation of anomalous resistance is usually assumed to be
𝑅 = 𝑅0 [𝑒−𝑡
𝑡2 − 𝑒−𝑡
𝑡1 ] where t1 is the characteristic rising time of anomalous resistance, t2 is the
characteristic fall time of anomalous resistance, and R0 is in the order of 1 Ω. In the Lee model it
is assumed that the plasma resistance becomes anomalous after the pinch phase.
Lee also developed his model to predict the ion numbers produced per shot from a plasma
focus [72]. In this research, the predicted ion number will be examined experimentally.
45
2.3 Lee Code
Lee uses an excel code to run his model. To run the Lee code, the required input parameters
are the capacitance (C0) , static inductance(L0), the geometry of electrodes (a, b), and filling
pressure(P0 ) of operating gas. The code is use to reproduce the experimental discharge current
and tube voltage as can be seen in Fig. 2.6.
The output of the code based on the simulated electrical signals of a plasma focus (i.e.
discharge current and tube voltage) are the plasma dynamics in the axial and compression phases,
plasma temperature and density, the number of neutrons generated per shot, soft x-ray yield, and
the ion beam yield. All the output results are plotted separately in an excel sheet. The parameters
have been used to produce the current and voltage are a= 1.5 cm, b= 4.5 cm, gas pressure p=3
Torr, C0= 4.5 µF, and V0= 30 kV.
Fig. 2.6: The discharge current and voltage waveforms simulated using Lee code.
46
Chapter 3
DPF UOFS-1
3.1 Introduction
The design and fabrication of any dense plasma focus machine can generally be divided
into two main steps. The first step is the simulation and fitting process to find the best configuration
of the coaxial discharge tube composed of the cathode and anode. The important parameters in
this step are the characteristics of the capacitor bank and the types and pressure of the filling gas.
The second step involves fabrication and assembly of the vacuum chamber, spark gap, trigger and
control circuits, discharge tube, and transmission line and diagnostic tools.
The type of dense plasma focus can be chosen by the available bank energy. Many reports
presented an efficient operation of a Mather-type plasma focus with a low stored energy. In this
thesis, a low energy capacitor bank, Mather-type plasma focus has been implemented. The aim of
the optimization of the discharge tube is to get an achievable compression of the plasma near the
peak discharge current of the capacitor bank. To reach this goal, two key parameters are important.
The first parameter is the energy density, which is expressed as 𝐸
𝑎3 , and the second is the drive
parameter 𝐼
𝑎𝑝0.5 , where E, a, I and p are stored energy, anode radius, peak discharge current, and
pressure, respectively. The energy density and the drive parameter is in the same range in all
Mather-type plasma focus devices, almost independent of the stored energy. In all DPF machines
with different stored energies, the drive parameter is in the range of 60 and 120 kA
cm ∙ mbar 0.5 and
the energy density is in the range of (1 − 10) × 1010 kJ
cm3 in deuterium gas.
47
3.2 Optimization of DPF UofS-I as an Ion Source
After considering the static inductance of the capacitor and transmission line used in the
experiment(L0), the maximum stored energy of the device (E≈2 kJ), the estimated peak discharge
current (I) based on magnetic and electrostatics energy balance ( E ≈ L0 I2) , the aforementioned
scaling parameters have suggested a feasible range of anode radius between 0.8cm and 1.7cm. To
optimize the drive parameter of a DPF machine as an ion source for a fixed stored energy, the
operating pressure, and plasma sheath velocity must be considered. The operating pressure plays
an important role in the beam loss. Plasma sheath velocity is important as it is related to the pinch
formation and the induced diode voltage. Taking these aspects into account in the optimization
process for deuterium gas (explained in the following parts), the drive parameter has been chosen
to be 90 kA
cm ∙mbar0.5 (𝑎 = 1.5 cm, 𝑝 = 1 Torr , and Ipeak =140 kA) for the DPF UofS-I machine.
To produce the positron emitting isotopes, one of the possible medical applications of a
DPF device, two main parameters are crucial for the ion beam production. The first is the energy
of ions, and the second is the ion beam current. The higher energy and number of the ions lead to
a higher rate of isotope production. It should be noted that the minimum ion energy to produce
nitrogen-13 using deuteron beam is 340 keV.
The energy of the ion beam and the ion beam current generated in DPF devices are
governed by the discharge current, plasma sheath velocity, gas pressure, anode shape and circuit
parameters. The higher discharge current and better optimization of the anode shape generate more
energetic ions, and the lower operating pressure beam leads to a lower beam loss, or a higher ion
48
number per shot. Plasma sheath velocity is another important optimization parameter for a DPF
device.
In order to optimize the parameters of the DPF UofS-I device to maximize the isotope
production, the Lee model has been used. The Lee model gives good estimations on the total
number of generated ions, the beam energy, and the beam current, which are important parameters
for isotope production.
The simulation results show that the total number of generated ions, beam energy and
current are maximized with the following parameters: the anode radius a=1.5 cm, anode length
b=7 cm, and the deuterium gas at working pressure of 1mbar. By choosing the maximum radius
of the anode, the best focusing effect can be achieved under low operating pressures, leading to a
higher sheath velocity and minimum beam loss effect in the machine. Maximum anode radius
corresponds to a lower inductance of discharge tube and a higher discharge current. According to
previous studies on ion beam emission from DPF machines (operating in deuterium gas) the
highest sheath velocity of 10 cm/µs is optimum for good focusing effect and production of high
energetic deuterons. This velocity is limited by a suitable plasma sheath thickness for strong
focusing effect. The DPF UofS-I simulation results show a peak discharge current that can reach
up to 140 kA, as can be seen in Fig. 3.1. The current drop shows the transfer of energy to plasma
during the pinch. The Lee model also shows a lifetime of focused plasma about 20 nsec.
49
Fig. 3.1: The peak tube current approximately 140 kA and tube voltage reaches up to 30 kV. The
operating pressure is 1 Torr.
The peak axial velocity about 10 cm/µsec can be achieved under the operation conditions
assumed in the simulation code, as can be seen in Fig. 3.2.
Fig. 3.2: The peak axial velocity about 10 cm/µsec under the operation conditions.
50
The simulation also shows that, in the pinch phase, the plasma density can reach
1017 ions/cm3, and that the plasma temperature is about 1 keV, as can be seen in Fig. 3.3. The Lee
model predicts: 1015 ions per shot, beam energy of 12 J, and beam current of about 15 kA, for the
optimized operating condition of the device; the amount of energy that is transferred to the pinch
is about 10 percent of the stored energy, and less than 10 percent of the stored energy in the focused
plasma will be converted to ion beam energy, making 1% energy efficiency for ion beam
production.
The parameters of the DPF UofS-I are listed in Table 3.1. It should be noted that the
insulator material and length are two important parameters to be optimized experimentally in the
device. In the DPF UofS-I, a Pyrex tube with 2 mm thickness is used as an insulator. The insulator
length can be changed up to 7 cm to cover the whole anode length.
51
Fig. 3.3: The simulated plasma temperature and ion density in DPF UofS –I.
52
Table 3.1: The parameters of DPF UofS-I.
Value Parameter
F 4.9 Capacitance (C0)
150 nH Inductance (L0)
18-30 kV Charging voltage (V0)
kJ 2 Stored energy (E0)
140 kA Peak discharge current
1.5 cm Anode radius (a)
5 cm Cathode radius (b)
7 cm Anode length (L)
3 cm Insulator length (d)
Pyrex Insulator material
Copper Electrodes material
rods Cathode type
12 Number of rods
, Ar2, N2H Operating gases
0.1 to 5 Torr Operating pressure
3.3 Experimental Setup of DPF UofS-I
Fig. 3.4 shows the DPF UofS-I with a stored energy of 2 kJ that is built at the University
of Saskatchewan. The DPF-UofS-I consists of a discharge tube and a vacuum chamber, a spark
gap, a capacitor bank, a 30 kV power supply, and a trigger circuit. A Faraday cage was used for
the oscilloscopes to record signals so the noises are minimized.
53
Fig. 3.4: The DPF UofS-I that was built in the University of Saskatchewan.
The schematic of the electrode assembly and the vacuum chamber of the DPF UofS-I can
be seen in Fig. 3.5. The Faraday cup was placed on the top of the anode. The soft x-ray detector
was placed on the side of the chamber to face the plasma column. A pinhole is placed in the
opposite side of soft x-ray detector for visible imaging. Two ports were used for pumping and gas
inlet in the chamber.
54
Fig. 3.5: The assembly of electrodes and detectors on vacuum chamber.
The discharge electrodes include 12 cathode bars surrounding the anode electrode. A knife
edge (to form a uniform discharge around the insulator) was made on a copper plate for uniform
formation of current in the break-down phase. The electrodes are made from copper and are
isolated electrically by a Pyrex tube, as can be seen in Fig. 3.6.
55
Fig. 3.6: Electrodes and insulator configuration after thousands of shots inside the vacuum
chamber.
A spark gap switch operating in high pressure was made to carry current from the capacitor
to the discharge tube. The electrodes were designed in such a way that the gap between them can
be changed. An SF6 gas and its pressure can be used to control the breakdown voltage of the spark
gap. The electrodes are made of copper, and the housing of spark gap is made by Delrin plastic
material to insulate the anode and cathode. A cylindrical plate was used as the trigger electrode. A
negative pulse with a peak voltage up to 40 kV was used to trigger the spark gap. A high voltage
transformer is used for electrical isolation of pulse generator from the capacitor bank. The
schematic of the spark gap used in the DPF UofS-I can be seen in Fig. 3.7. The spark gap is placed
in a symmetric configuration made by 8 cathode rods to reduce the inductance of the transmission
line. The coaxial transmission line also reduces the discharge noise during the operation of the
device. The capacitor and transmission lines were placed inside a grounded shielding box for safety
issues, as can be seen in Fig. 3.4.
56
Fig. 3.7: The schematic of spark gap used in DPF UofS-I. The bottom part of e spark is connected
to capacitor while the upper part of spark gap is connected to chamber.
For charging the capacitor, a 100 Ω charging resistor, along with a power supply, were
used. The voltage can be set on the power supply, and the capacitor discharge through a spark gap
by manual or automatic trigger switch is connected through an optical fiber to a trigger box.
Discharging the capacitor bank can also been done manually by a high voltage relay connected to
a mega-ohm resistance. The charging time of the device is about 30seconds, and the damping time
-40 kV
Trigger
57
of the capacitor is about RC≃5 min through the dump resistor. The high voltage relay and the
damping resistance were tested up to 30 kV.
To evacuate the chamber, a two-stage rotary pump has been used to attain a low, ultimate
pressure of 10-3 mbar. A vacuum gauge was used to monitor the background and filling pressures.
All signals from the device were transmitted through 50 Ω coaxial cables to the Faraday
cage. Three Tektronix oscilloscopes (model 2014 c) with USB ports were used to record all signals
simultaneously. All oscilloscopes were trigged by a discharge trigger signal. To operate the DPF
UofS-I, the following steps are required:
1. Pumping the vacuum chamber by a rotary pump up to 10-3 Torr.
2. Filling the vacuum chamber with gas up to operating pressure.
3. Charging the capacitor bank.
4. Trigging the spark gap to discharge the capacitor.
5. Closing the discharge relay to clear the residual charge in the capacitor.
A Faraday cage that was set up in the plasma focus lab is used for the shielding of
electromagnetic pulse (EMP) generated during the capacitor discharge. The Faraday cage was
made of copper mesh and was grounded to effectively reduce the electromagnetic noise level.
Copper is a good conductor that absorbs high frequency electromagnetic noise. To prevent a
ground loop, the Faraday cage is connected to the DPF device on a single point.
Finding the optimum experimental charging voltage and filling pressure of the device to
produce a strong focusing is the most important step for running a plasma focus device. Changing
the operating gas can change the operating charging voltage and pressure. As mentioned earlier,
58
argon, nitrogen, and hydrogen are used as the operating gases. The operating conditions of the
DPF UofS-I in different gases can be seen in Table 3.2.
Table 3.2: The operating conditions of DPF UofS-I.
Operating gas Charging voltage (kV) Operating pressure (Torr)
Argon 15- 28 0.1 - 0.5
Nitrogen 15-28 0.1 - 0.7
Hydrogen 15-28 1 - 6
It should be noted that operating the DPF UofS-I in light and heavy gases is a unique
feature of the device as it is a low energy device.
Fig. 3.8 shows the discharge current (I) and anode voltage (V-Anode) of DPF UofS-I
operating in argon gas. The drop of the current and the peak of the anode voltage shows the
pinching effect. As can be seen in Fig. 3.8, the duration of the current drop is about 500 ns, and
anode voltage shows two main peaks. The discharge current shows a 50 percent drop during the
final stage of the plasma focus, indicating a very strong pinch result.
59
Fig. 3.8: The focusing effect of DPF UofS-I operating in argon gas. V =28 kV, P=0.1 Torr.
V-a
no
de
(kV
)
t(µs)
60
Chapter 4
DPF UOFS-1 DIAGNOSTICS
4.1 Introduction
In order to study the time-resolved signals of the charged particles and x-rays from the
pinching plasma, different detectors, such as an ion beam detector, electron beam detector, a soft
x-ray detector, and hard x-ray detector, had to be developed. The main goal in developing these
detectors is to measure the flux (number per shot) and the energy spectrum of the ions, the energy
spectrum of radiated soft and hard x-rays, and electron beam current. As mentioned in the previous
Chapter, the DPF UofS-I is developed as an ion source. In order to find the best operating condition
of this device to produce energetic ions, the combination of the ion beam, electron beam, and x-
ray detectors are also required. Plasma temperature is another important parameter to be measured
during the compression phase. The measurement of plasma temperature can be done by soft x-ray
spectroscopy.
Another important part, regarding the ion beam production, is related to the optimum
operating parameters to increase the flux and energy of the charged particle beams from the device,
which can be understood by considering the correlation of the energetic ion beam emission with
the x-ray and electron beam emission from the device. In this chapter, charged particles and x-ray
diagnostics that were developed and used in the DPF UofS-I will be described, and the technical
details will be presented.
61
4.1.1 Discharge Current and Anode Voltage Probes
In general, to measure the discharge current derivative, a fast Rogowski coil can be used
in a plasma focus. The current that pass through the coil induces a voltage across the coil, as can
be seen in Fig. 4.1. Rc and Lc are the resistance and inductance of the coil, respectively. R is the
external resistance that is connected to the coil. Rogowski coils is a solenoid tightly wound on a
flexible form so it can be bent to let the two ends meet. Rogowski coils can be either work as a
self-integrator, or can be connected to a proper external integrator to measure the current. When a
Rogowski coil is used as a self-integrator, the output of the coil is proportional to discharge current.
Fig. 4.1: Rogowski coil equivalent circuit.
The voltage across the coil can be presented by Eq. 4-1, where I is the discharge current and i is
the current passing through the Rogowski coil.
62
dt
dIkiRR
dt
diL cc )( (4-1)
Eq. 4-1 shows that if ( )c c
diL R R i
dt , the output of the coil is given by Eq. 4.2 where
the Rogowski coil works as a self-integrator coil.
out
c
kV Ri R I
L
(4-2)
If dt
diLiRR cc )( , then the output of the coil can be shown by Eq. 4-3, where an
external integrator is needed for measuring the discharge current.
dt
dI
RR
RkiRV
c
out
(4-3)
In DPF UofS-I, a commercial Pearson fast current probe (with a response time less than
1 ns) was used to measure the discharge current. The current probe was connected to the ground
side and is insulated from the high voltage.
To measure the anode voltage, a fast Tektronix high voltage probe (P6015-A) with a
maximum operating voltage of 40 kV and response time of 2 ns was connected to the anode. The
current probe and high voltage probe can be seen in Fig. 4.2.
63
Fig. 4.2: The current probe and high voltage probe were used in DPF UofS-I.
4.1.2 Faraday Cup
One of the first detectors that has been built is a Faraday cup, which works in the secondary
electron emission (SEE) mode. The cup has been made for collecting ion beams, and has been
designed and biased for minimum possible noise ratio and good impedance matching. It should be
noted that the Faraday cup is a useful detector to measure the ion beam flux. In order to measure
the energy of the ion beam based on the time of flight measurement, a configuration with a Faraday
cup and an x-ray detector are required.
Based on the time-of-flight (TOF) method, the kinetic energy of ion beam can be estimated
by measuring the velocity and mass of a charged particle. The velocity is equal to the distance
between plasma region and the faraday cup position divided by the interval time between the peak
of the hard x-ray signal (onset of ion beam emission) and the peak of ion beam signal. Fig. 4.3
64
shows the Faraday cup assembly used in DPF-I. The Faraday cup was placed 75 cm away from
the pinching plasma. A pinhole with 500 microns in diameter was used to attenuate the ion beam
intensity before reaching the cup.
The inner cup was made of copper and an aluminum cage was used for EMP shielding.
The impedance of the cup based on the geometry of inner and outer cup was chosen to be matched
with the 50 Ω coaxial cable to avoid a pulse reflection.
Fig. 4.3: The Faraday cup assembly used in DPF-I.
The configuration of the Faraday cup can be seen in Fig. 4.4. The inner cup is negatively
biased to -1000 V. A cylindrical insulator (Teflon) was used to insulate the inner and the outer
cup.
65
Fig. 4.4: The configuration of Faraday cup was used to monitor the generated ion beam.
The typical signals of the ion beam current (IB) and the discharge current (I) can be seen
in Fig. 4.5. The duration of the beam is about 100 ns.
Fig. 4.5: Ion beam current (IB) and discharge current (I) signals.
To measure the ion current, the Faraday cup signal has to be converted from a voltage
signal to a current signal. The output signal of the cup is the voltage across a 50 Ω resistance. The
Ion To biased circuit Collector
t(µs)
66
current signal (IB) can be expressed by 𝐼𝐵 =𝑑𝑄𝑇𝑜𝑡𝑎𝑙
𝑑𝑡, where QTotal is the charge collected by the
Faraday cup.
The total collected charge by the cup can be presented by Eq. 4-4, where dq is the number
of the secondary electrons that is produced by the interaction of energetic ions by the inner cup, k
is the escape factor of a secondary electron from the inner cup, and dQ is a total charge of the ions.
The escape factor is equal to the ratio of the radius to the length of the collector. The ion collector
is usually negatively biased to push the secondary electron out of the cup.
𝑑𝑄𝑇𝑜𝑡𝑎𝑙 = 𝑑𝑄 + 𝑘𝑑𝑞 (4-4)
The number of secondary electrons (g) produced by the interaction of the energetic ion
with the collector can be calculated based on the energy of the ion and the incident angle of the
ion with respect to the normal of the collector surface. The coefficient g is called the electron
stopping power, and can be expressed by Eq. 4-5, by considering the electron stopping power as
(dE/dx)e and as the incident angle of ion.
edx
dEg
cos (4-5)
The coefficient, , depends on the material of the collector. The value of given for
copper is 0.22 Å/eV, based on the SRIM software.
By considering the charge and energy of the incident ion (Z(E)), kdq can be expressed by
Eq. 4.6.
𝑘𝑑𝑞 = 𝑑𝑄 ( 𝑘 𝑔
𝑍(𝐸) ) (4-6)
67
By considering dQ = dN e Z(E) where e is charge of electron, and dN is the number of ions
with an energy between E and E+dE, dQTotal can be expressed by Eq. 4-7.
𝑑𝑄𝑇𝑜𝑡𝑎𝑙 = 𝑑𝑄 (1 + 𝑘 𝑔
𝑍(𝐸) ) (1 +
𝑘 𝑔
𝑍(𝐸) ) = 𝑑𝑁 𝑒 𝑍( 𝐸) (1 +
𝑘 𝑔
𝑍(𝐸) ) (4-7)
By using Eq. 4-6, the total current that is measured by the cup can be expressed by Eq. 4-8
𝐼 =𝑑𝑄𝑇𝑜𝑡𝑎𝑙
𝑑𝑡=
𝑑𝑁
𝑑𝑡 𝑒 𝑍( 𝐸) (1 +
𝑘 𝑔
𝑍(𝐸) ) (4-8)
By using the chain rule 𝑑𝑁
𝑑𝑡=
𝑑𝑁
𝑑𝐸 𝑑𝐸
𝑑𝑡 ,𝐸𝐼𝑜𝑛 =
1
2𝑚 × 𝑣2, and 𝑣 = 𝐿/𝑡 where L is distance
between the ion collector and pinching plasma , Eq. 4-8 gives
𝐼 =𝑑𝑄𝑇𝑜𝑡𝑎𝑙
𝑑𝑡=
(2𝐸)32
𝐿 𝑀12
𝑒 𝑍( 𝐸) (1 + 𝑘 𝑔
𝑍(𝐸) )
𝑑𝑁(𝐸)
𝑑𝐸 (4-9)
Eq. 4-9 can be used to find 𝑑𝑁(𝐸)
𝑑𝐸 versus the ion energy by using the IB signal, the mass
of ion, the charge of ion, and the distance of the Faraday cup from the pinch. To use Eq. 4-9 to
plot 𝑑𝑁(𝐸)
𝑑𝐸 -E one needs to convert IB –T to IB-E. In the following chapter, the IB-E plot will be
presented for different operating gases.
Optimization of a Faraday cup can be done by biasing the voltage, adding a magnetic field,
and reducing the photoelectric effect. The beam loss effect due to the interaction of the ion beam
with the background gas is another issue that can be resolved by reducing the pressure inside drift
tube. It should be noted that measuring the exact number of ions generated from the pinching
plasma is an unsolved technical problem.
68
4.1.3 Electron Beam Measurement
A plasma focus is an energetic electron beam source. The electron beam current can reach
to several kA, and the energy of the electrons can reach up to MeV. The duration of the electron
beam generation is on the order of 100 ns in a plasma focus. Due to the generation of the electron
beam in a short period of time, a fast Rogowski coil is required to monitor the electron beam
current.
To extract the electron beam from plasma focus, a hollow anode has to be used. The hollow
anode has to be connected to the drift tube and a current probe, which ensure the electron beam
current to pass through the tube. A Pyrex tube was used to extract the electron beam from the
bottom of the hollow anode. The distance between the collector and pinch is about 30 cm. To avoid
a discharge of the capacitor bank through electron beam collector, a 5 Ω resistance has been used
in the ground side of the collector. A Faraday cup can be biased negatively to measure the energy
of the electron. An electron with an energy less than the biased voltage cannot reach the cup.
A self-biased Faraday cup is another method to measure the electron energy. In this
method, a resistance R can cause a negative voltage by passing the electron current through the
resistance (i.e. V=R IE-beam). Adding filters to block the low energy electrons is another method to
distinguish the low energy and high energy electrons generated from the pinching plasma. The
energy spectrum of the radiated hard x-ray can also give a good estimation of the energy of the
generated electron by the pinching plasma. A correlation of the ion and electron beams can give
an idea about possible mechanisms of electron beam generation from the pinching plasma.
69
Figure 4.6 shows the typical signal of an electron beam (I-E-beam) generated from the
DPF UofS-I. The duration of the beam is about 100 ns. The electron current can reach up to 2 kA
in DPF UofS-I. It should be noted that DPF UofS-I is a unique low energy device in terms of
having these charged particle detectors.
Fig. 4.6: A typical signals of electron beam (I-E-beam) generated from DPF UofS-I.
4.1.4 Hard X-ray Measurement
In a plasma focus device, hard x-rays are generated due to bombardment of the anode tip
by energetic electrons. The duration of hard x-ray radiation in a plasma focus is about 100 ns. Hard
x-rays can be detected by a plastic scintillator in front of a photo-multiplayer tube (PMT). Two
hard x-ray detectors were made in this project to detect the generated x-rays from the device. The
t(µs)
70
detectors were placed inside a Faraday cage three meters away from the pinching plasma. A 1cm
stainless thick steel plate was used to filter the hard x-rays.
The PMT tube can be biased negatively up 1600 V. Increasing the biased voltage will
intensify the signal of the hard x-rays. The delay time between the hard x-ray detectors is about
30 ns measured in the lab. Figure 4.7 shows the typical signal of the hard x-ray from DPF UofS-I.
Fig. 4.7: The typical signals of hard x-ray from DPF UofS-I.
4.1.5 Soft X-ray Measurement
The definition of soft and hard x-rays is based on the energies of the x-ray. In general, in a
plasma focus, the x-ray energy less than 4 keV is considered as soft x-ray. The energy spectrum
of the radiated x-ray from the pinching plasma gives an idea about the energy of an electron and
stage of ionization of atoms inside the plasma column. The number of ionization of atoms inside
71
the plasma column gives an estimation of the plasma temperature during the compression of the
plasma.
To measure the energy spectrum of the radiated x-rays from DPF UofS-I, two diode based
x-ray spectrometers were developed. Aluminum, cobalt, titanium, and beryllium filters, with
different thickness, were used to measure the energy spectrum of the x-ray. The proper filters can
be chosen by considering the operating gas.
To build the diode based x-ray spectrometer, four fast photodiodes (BPX-65) with a
response time of less than 1 ns were biased negatively to about -50 volts. Figure 4.8 shows the
biasing circuit of the BPX-65 diode. A 10 pF capacitor has been used to filter the noise.
Fig. 4.8: The biasing circuit of BPX-65 diode.
72
The radiated x-rays with energy less than 30 keV can be absorbed by the active layer of
the diode. Absorption of x-rays by diodes produces electron holes (i.e. empty place of an electron)
that can carry current. It should be noted that the BPX-65 photodiode response is almost constant
to the energy-spectrum of soft x-rays between 0.5 keV to 4 keV, and reaches its maximum in that
same range. Therefore, the output of diodes can give an estimation of the number of photons that
is absorbed by a diode in that range. Figure 4.9 shows the sensitivity of the BPX-65 diode to the
energy spectrum of x-rays between 0.5 keV to 4 keV (i.e. 0.3 nm < λ< 2 nm).
Fig. 4.9: The sensitivity of BPX-65 respect to energy spectrum of x-ray.
In a plasma focus, the copper K-lines have a significant contribution to the radiated x-rays
from the pinching plasma. The energy of the copper K-lines is about 8keV. The sensitivity of the
73
BPX-65 diode is about 0.02 C/J in that range, which is not significant compared to the sensitivity
of the BPX 65 diode in the energy range of 0.5 keV to 4 keV, as can be seen in Fig. 4.9.
The total sensitivity of the detector can be calculated based on the transmission of the
filters, the gas, and the sensitivity of the diode. To convert the signal from the diode to energy, the
following steps have to be considered.
1. The voltage from the oscilloscope should be converted to current by dividing the
voltage by 50 Ω output resistance.
2. The total charge has to be calculated by integration of the current over time.
3. The total charge should be divided by the sensitivity of diode in the particular
wavelength that is absorbed by the diode.
The area of the diode is about 1 mm2. The total energy of the radiated x-ray from the
pinching plasma can be calculated by Eq. 4-10, where r is diode distance from the pinch in mm.
The distance of diodes from the pinch is about 40 cm.
E Total = E Absorbed by diode × 4 π r 2 (4-10)
The two soft x-ray detectors with 8 active channels were connected to the chamber on the
top and on the side of the chamber. The detectors were isolated from the chamber by a ceramic
break to break the ground loop and to reduce the noise from the pinch. Fig. 4.10 shows the two
soft x-ray detectors that were connected to the chamber. Numbers 1 and 2 show the side and top
side detectors, respectively.
74
Fig. 4.10: The two soft x-ray detectors were used in DPF UofS-I.
The typical signal of the soft x-ray detector from one channel of soft x-ray detectors can
be seen in Fig. 4.11. The duration of the x-ray signal is about 300 ns.
Fig. 4.11: The soft x-ray signal from DPF UofS-I.
75
Moreover, the plasma temperature measurement based on the ratio method is one of the
significant data that has been obtained by soft x-ray spectroscopy in DPF UofS-I.
4.1.5 Signal Recording System
All signals from the ion beam detector, electron beam detector, hard and soft x-ray
detectors, along with the discharge current and anode voltage, were recorded with three
oscilloscopes. The three digital storage oscilloscopes (Tektronix, 2014 C model) were triggered at
the same time at the start of the discharge current.
All signals were stored in a USB connected to each oscilloscope in excel format. A
standard 50 Ω coaxial cable was used to transfer signals from the sensors on the device to the
Faraday cage.
76
Chapter 5
EXPERIMENTAL RESULTS AND
DISCUSSION
5.1 Introduction
In this chapter, the experimental results on charged particle emission and x-ray radiation
will be presented. In the first part, the effect of the atomic number of gas on x-ray radiation and
ion beam emission will be presented. Based on the measured plasma inductance, the effect of the
atomic number of gas on the compression of plasma will also be discussed. The measured ion
energy and ion current density will be compared with the plasma voltage. The ion current density
generated from the pinching plasma will be compared with the Lee model.
In the second part, by analyzing the electrical signals, a new method to estimate the plasma
resistance in a plasma focus device will be introduced. Anomalous plasma resistance is one of the
main mechanisms of the energetic charged particle generation from a pinching plasma. The effect
of the atomic number of the gas on the occurrence of anomalous resistance will be discussed. To
measure the plasma resistance, the plasma voltage and inductance will be derived by analyzing the
waveforms of the discharge current and the anode voltage of the device.
In the third part, the required electric field to generate runaway charged particles from the
pinching plasma will be discussed. The Dreicer theory and the avalanche process will be discussed.
By measuring the plasma voltage and inductance, the electric field across the plasma column for
77
runaway charged particle generation will be calculated and compared with the Driecer electric
field. The time rate of the runaway electron generation will be calculated in a plasma focus. The
effect of anomalous heating on a required electric field for runaway charged particle production
will be studied experimentally. By using soft x-ray spectroscopy, anomalous heating in the post
pinch-phase will also be shown experimentally.
In the last part of this chapter, a new type of a DPF device with semi-active electrodes
(DPF-SAE) will be introduced. Technical details of an SAE-type plasma focus will be presented.
The electrical discharge signals of the device will be analyzed. The generation of charged particles
and x-rays from the device will be investigated. The results corresponding to the charged particles
and x-ray radiations from a DPF-SAE and DPF UofS-I will be compared.
5.2 Charged Particle Emission in Light and Heavy Gases
In this experiment, the generation of charged particles and x-rays from argon, nitrogen,
and hydrogen plasma will be presented. Different observed regimes of the focusing in the operating
gases will be discussed. The energy spectrum of the ion beam in three operating gases will be
presented and compared with the measured plasma voltage. The change in plasma inductance will
be measured by analyzing the discharge current and anode voltage.
Fig. 5.1 shows the different regimes of the focusing exhibited in the typical signals of the
discharge current and high voltage probe signals in argon and hydrogen gases. As can be seen, the
duration of the current drop and the peak of anode voltage in argon gas is about 400 ns. In the case
of hydrogen, the duration of the discharge current and the anode voltage is about 100 ns. These
features have been observed in all operating pressures of the device in argon and hydrogen gases.
The long duration of x-rays and ion beams have been observed in argon gas, and have been
78
compared to the hydrogen and Nitrogen gases. Table 5.1 shows the duration of ion beams and x-
rays generated from the DPF UofS-I in the three operating gasses.
Fig.5.1: The different regimes of focusing in argon and hydrogen gasses.
V-a
no
de
(kV
)
0
20
40
60
80
100
120
140
0
5
10
15
20
25
30
35
1.5 1.55 1.6 1.65 1.7 1.75 1.8
I(k
A)
V-a
nod
e(k
V)
t(µs)
Hydrogen gas
V-a
no
de
(kV
)
t(µs)
V-a
no
de
(kV
)
79
Table 5.1: The main features of the experimental observations for plasma foci operating
in argon and hydrogen gasses.
The waveforms of the discharge current and anode voltage of the DPF UofS-I can
interpreted by a circuit analyses [73-78]. The tube voltage can be calculated through the discharge
current and anode voltage, based on Eq. 5-1, where 𝑉𝐴(𝑡) is the anode voltage, 𝐼(𝑡) is the
discharge current, RPl is the plasma resistance, and 𝐿0 is the inductance of the discharge circuit
from the voltage probe to the discharge electrodes. Considering that 𝐿0 = 40 𝑛𝐻, the tube voltage
can be expressed by Eq.5-1.
𝑉𝑇𝑢𝑏𝑒(𝑡) ≃ 𝐼𝑅𝑃𝑙 +𝑑
𝑑𝑡[𝐿𝑡𝑢𝑏𝑒 (𝑡)𝐼(𝑡)] ≃ 𝑉𝐴(𝑡) − ( 𝐿0)
𝑑𝐼
𝑑𝑡 (5-1)
The power into the discharge tube can be calculated by 𝑉𝑇𝑢𝑏𝑒(𝑡)𝐼(𝑡). Figure 5.2 shows the
tube voltage in argon gas corresponding to the anode voltage and the discharge current in Fig. 5.1.
Numbers 1 and 2 correspond to the pinch and post-pinch phases.
Operating GasGas Hydrogen Nitrogen Argon
Operating Pressure (Torr) 4 – 6 0.1-0.6 0.1- 0.5
dI/dt (A/s) ≤8×1011 ≤6×1011 ≤1.2×1012
Current drop (kA) ≤ 15 ≤ 30 ≤ 60
Anode Voltage (kV) ≤ 30 ≤ 30 ≤ 45
Current drop duration (ns) ≤ 50 ≤200 ≤ 300
Hard x-ray duration(ns) ≤ 20 ≤ 50 ≤150
Charged Particles duration (ns) ≤ 100 ≤ 200 ≤ 250
Soft x-ray duration(ns) Noise level ≤150 ≤300
80
Fig. 5.2: Discharge current and tube voltage in argon gas during the compression phase.
Since the inductance of the electrode is about 12 nH, the voltage across the plasma in the
compression phase can be calculated by assuming L0 = 52 nH (i.e. inductance of the discharge
circuit from the voltage probe to the tip of the discharge electrodes). The estimated voltage across
the plasma column during the pinching time reaches up to 65 kV in hydrogen, 57 kV in nitrogen
and up to 100 kV in the argon plasma.
The tube inductance can be calculated by measuring the tube voltage and the discharge
current by Eq.5-2, where 𝑡0 is the start time of the discharge.
𝐿𝑇𝑢𝑏𝑒(𝑡) =∫ [𝑉𝑇𝑢𝑏𝑒(𝑡)−𝐼(𝑡)𝑅𝑃𝑙(𝑡)]𝑑𝑡 +
𝑡𝑡0
𝐿𝑇𝑢𝑏𝑒(𝑡𝑐) 𝐼(𝑡𝑐)
𝐼(𝑡) (5-2)
Fig. 5.3 shows the typical measured change in tube inductance in hydrogen gas. The rapid
change in tube inductance, which is correlated with the first peak in the plasma voltage, has been
considered as the change in plasma inductance during the pinch phase. As can be seen in Fig. 5.3,
81
the change in the tube inductance during the compression phase can reach up to 40 nH in argon
gas, while in hydrogen and nitrogen gases is about 10 nH.
Fig. 5.3: Change in tube inductance in hydrogen gas.
The significant change in plasma inductance in argon gas compared to the nitrogen and
hydrogen gases can explain the radiation-enhanced compression in high-z gases. The theoretical
study of radiation-enhanced compression of a DPF by Braginski and Pease shows the feasibility
of radiation cooling of plasma during the pinch phase [79]. Intense radiation of x-ray leads to
decreasing the static pressure of the plasma. Decreasing the static pressure of the plasma causes a
radiation-enhanced compression of the plasma column in the pinch phase. The increased
compression of the plasma is eventually terminated by the plasma self-absorption as the plasma
becomes highly dense.
The Lee model was used in argon and hydrogen gases to simulate the dynamics of the
plasma in the compression phase. The simulated current waveform is fitted to the measured current
waveform based on electrodynamics and thermodynamics of the measured situation, both in the
t(µs)
82
axial and the radial phases, for particular shots, as shown in Fig. 5.4. The simulation results show
the radial and axial trajectories of the plasma during the pinch phase in both (a) hydrogen, and (b)
argon gases, as presented in Fig. 5.4.
The minimum ratio of the pinch plasma radius to the anode radius in the hydrogen gas is
0.14, compared to the argon with a very small value of 0.012. Based on the minimum radius of the
plasma column, the change in plasma inductance is about 8 nH in the hydrogen gas, and is about
40 nH in the argon gas. These results are in agreement with the measured results shown in
Fig. 5.4, and could explain the measured large inductance of the argon pinch. The smaller radius
of the plasma in argon gas can be explained by the plasma energy gain/loss during the compression
phase. Based on the simulation results, the pinching plasma in argon gas loses 60 J over a radiative
period of around 15 ns. During the most intense period, 30 J is radiated in FWHM time of 4 ns. In
an argon pinch plasma, the thermal energy stored at the start of the radiation is computed to be
200 J. Therefore, 30% of the pinch thermal energy is radiated away during the pinch time. This
level of radiation is sufficient in argon to fulfill the condition of energy depletion for strong
radiative-enhanced compression of the pinching plasma. The analysis for the hydrogen shot shows
that the loss of energy due to the x-ray radiation is only a small fraction during a pinch. Thus, the
x-ray radiation from the hydrogen pinching plasma is not significant to cause a reduction of the
pinch radius.
83
Fig. 5.4: The simulated radial and axial trajectories of plasma in (a) hydrogen gas, and
(b) argon gas.
The power into the discharge tube can be calculated using𝑉𝑇𝑢𝑏𝑒(𝑡)𝐼(𝑡). Fig. 5.5 shows the
power into the plasma in argon gas, corresponding to the anode voltage and discharge current, in
Fig. 5.1. Numbers 1 and 2 correspond to the pinch and post pinch phases.
(a)
r min
rmin
(b)
84
Fig. 5.5: Power into plasma in argon gas during the compression phase.
Based on the measured voltage across the plasma in the compression phase, and power
into plasma, integrating the power over time during the compression phase gives the total energy
injected into plasma in argon gas. This gives about 500 J in argon gas, 250 J in nitrogen gas, and
200 J in hydrogen gas. It is interesting to note that simulating the long drop of the discharge current
shown in Fig. 5.1, based on Lee model for argon gas, requires anomalous resistance to explain the
significant injected energy into plasma after the pinch duration [45, 12, 73]. In the following parts,
the simulated results corresponding to the plasma anomalous resistance will be analyzed by circuit
analysis of the device.
The effect of the atomic number of gas on the energy spectrum of ions generated from the
pinching plasma was studied by measuring the ion beam energy based on the time of flight method.
To measure the ion energy, the peak of the anode voltage (peak 1) is considered as the start of the
ion emission. Peak 2 shows the hard x-ray signal. As can be seen in Fig. 5.6, the delay time between
V-anode and H-X traces is about 30 ns. Peak 3 shows the proton beam signal.
t(µs)
85
Fig. 5.6: Anode voltage, hard x-ray and ion beam signals in hydrogen gas.V0 =28 kV, P=5 Torr.
Based on the time of flight method, the typical energy spectrum of the proton generated
from the DPF UofS-I in maximum charging voltage and different pressures can be seen in
Fig. 5.7. Protons with energies between 15 keV to 20 keV contributed significantly in the peak of
the proton current density. In the case of nitrogen, the ions with energies between 80 keV to
140 keV contributed significantly in the peak of ion current density. In the case of argon, ions with
energies between 150 keV to 400 keV contributed significantly in the peak of the ion current
density.
1
2
3
V-a
nod
e (k
V)
t(µs)
86
Fig. 5.7: Energy spectrum of proton generated from DPF UofS-I in maximum operating
charging voltage.
The peak of ion current does not depend on the plasma voltage. However, increasing the
plasma voltage shifts the energy spectrum of the generated ions to higher values.
The Lee model was used to predict the ion current density in the operating gases and
pressures. A comparison of the experimental with the measured results in the three operating gases
can be seen in Fig. 5.8. The results show that the simulated and measured ion current density have
the same order of magnitudes in the operating gases and pressures.
Hydrogen ion beam energy
87
Fig. 5.8: The simulated and measured ion current densities in DPF UofS-I.
5.3 Anomalous Resistance and Injected Energy into Plasma
The inductive and resistive nature of the plasma impedance in a plasma is an unsolved
issue. The time variation of the plasma voltage during the compression phase may be interpreted
based on two scenarios. In the first scenario (inductive scenario), it is assumed that the plasma
resistance during the final stage of the plasma focus is negligible [74]. In the second scenario
(inductive-resistive scenario), it is assumed that plasma resistance becomes anomalous after the
pinch phase (t > t pinch), and that the inductive and resistive plasma voltage drops should be
considered simultaneously [28].
Experimental results
88
The amount of energy that is injected into the plasma in argon gas is more than that in
nitrogen and hydrogen gases, as experimentally obtained. To simulate the current trace and the
tube voltage in argon gas in the DPF UofS-I, anomalous resistance has to be taken into account in
the Lee model. Fig. 5.2 shows that the tube voltage, after the pinch (i.e. peak 2), is higher than the
tube voltage during the pinch (i.e. peak 1). Considering that the tube voltage is given by Eq. 5-1,
the difference between the tube voltages at time 2 and 1 gives Eq. 5-4
𝑉𝑇𝑢𝑏𝑒(𝑡2) − 𝑉𝑇𝑢𝑏𝑒(𝑡1) = (𝐼𝑅𝑃𝑙(𝑡2) − 𝐼𝑅𝑃𝑙(𝑡1)) + (5-4)
(𝑑𝐿𝑡𝑢𝑏𝑒 (𝑡2)
𝑑𝑡𝐼(𝑡2) −
𝑑𝐿𝑡𝑢𝑏𝑒 (𝑡1)
𝑑𝑡𝐼(𝑡1)) + (
𝑑𝐼(𝑡2)
𝑑𝑡𝐿𝑡𝑢𝑏𝑒 (𝑡2) −
𝑑𝐼(𝑡1)
𝑑𝑡𝐿𝑡𝑢𝑏𝑒 (𝑡1) )
The plasma resistance is negligible before the pinch (i.e. 𝐼𝑅𝑃𝑙(𝑡1) ≃ 0). The change in
plasma inductance reaches to its maximum during the pinch
(𝑑𝐿𝑡𝑢𝑏𝑒 (𝑡2)
𝑑𝑡𝐼(𝑡2) −
𝑑𝐿𝑡𝑢𝑏𝑒 (𝑡1)
𝑑𝑡𝐼(𝑡1) < 0). Because the time rate of change of current is negative, and
its magnitude at t1 is smaller than its magnitude at t2, as measured, then the 3rd term is also negative
for the case when the plasma inductance at t2 is larger than that at the earlier time t1 (i.e.
𝐿𝑡𝑢𝑏𝑒 (𝑡2) > 𝐿𝑡𝑢𝑏𝑒 (𝑡1) ). Even if 𝐿𝑡𝑢𝑏𝑒 (𝑡2) < 𝐿𝑡𝑢𝑏𝑒 (𝑡1) , the magnitude,
|𝑑𝐼(𝑡1)
𝑑𝑡𝐿𝑡𝑢𝑏𝑒 (𝑡1)| < 3 kv ( 𝐿𝑡𝑢𝑏𝑒 (𝑡1) ≈ 30 nH), and
𝑑𝐼(𝑡1)
𝑑𝑡 ≃10 -11 (A/s), By subtracting the tube
voltage at the time t2 from that at t1, Eq.5-4 gives
𝐼𝑅𝑃𝑙(𝑡2) > 7 kV (5-5)
0
<0
0 0
89
Considering that the discharge current is about 50 KA, 𝑅𝑃𝑙(𝑡2) ≃ 0.2 Ω, which is
considered as anomalous resistance [28, 73]. This result verifies the occurrence of anomalous
resistance in the post-pinch phase in argon gas as the classical resistance of plasma in the order of
1 mΩ.
The injected energy during the compression phase reaches its maximum in the argon
plasma. According to Fig. 5.5, the injected power into the tube after the pinch (i.e. peak 2) is
significant compared to the pinch phase (i.e. peak 1). The injected energy into the plasma in argon
focus after pinch is consumed by the anomalous resistance, as calculated by Eq. 5-5.
A significant increase of the plasma resistance at the post-pinch phase in the argon plasma
might be related to the occurrence of a lower hybrid drift instability (LHDI) in such a phase [80-
81]. The LHDI in the plasma is plausible when the drift velocity of the electrons (Vd) reaches the
ion sound speed (𝑉𝐼𝑜𝑛 ) (i.e. 𝑉𝑑
𝑉𝐼𝑜𝑛≥ 1 ). The ratio of Vd and Vion can be expressed by Eq. 5-6, where
Z is the effective charge of the ion, M is the ion mass, and N is linear plasma density [80-81].
𝑉𝑑
𝑉𝐼𝑜𝑛𝛼 √
(𝑍+1)𝑀
𝑁 𝑍3 (5-6)
Eq.5.6 shows that the ratio of Vd to the V ion depends on the density, the atomic mass, and
the charge of the ionized atom. The LHDI is more plausible when the plasma density drops, or if
the number of the ionization of the gas decreases. In a DPF, the plasma density decreases in the
post-pinch phase, as compared to the pinch phase. It is interesting to note that decreasing the
plasma temperature in the post-pinch phase, as compared to the pinch phase, can be followed by
decreasing the effective charge of the ion (Z) in heavy gasses. The combination of the drop of
plasma density and temperature can significantly increase the likelihood of the occurrence of
90
LHDI. This could explain the significant contribution of the anomalous resistance in the argon gas
than in the hydrogen and nitrogen gases.
Plasma heating by an anomalous resistance (caused by the LHDI) can be expressed as
PJ = (1+ 4 𝜒 ) Pclassic in a plasma focus, where Pclassic is the plasma heating by classical resistance,
and 𝜒 can be expressed by the following Eq.5-7 [80-81].
𝜒 = 7.95 × 1047 𝐼4 𝐴1/2 𝑎
𝑁7/2𝑍7/2(𝑍+1)1/2 (5-7)
Here, a is the plasma radius, and A is the ion mass number. Eq.5-7 shows that the drop of
the linear plasma density, and decreasing the atomic number of the gas, can significantly increase
the injected power into the plasma by the anomalous joule heating process. The anomalous joule
heating could terminate the radiation – enhanced compression in the argon gas.
5.4 Plasma Heating and the Emission of Runaway Charged Particle
The anomalous resistance can heat the plasma through the anomalous joule heating process. This
Section will discuss the effect of the anomalous joule heating on runaway charged particle
emission and x-ray generation during the post-pinch phase.
A test particle in a plasma becomes runaway when the decelerating force (imposed on the
test charge by other charge particles) becomes negligible compared to the accelerating force
exerted by the electric field. According to the Dreicer theory, this condition will be satisfied when
the accelerating force on a test charge becomes greater than the critical electric field, as defined in
Eq. 5.8 in a plasma, where n and T are the electron density and temperature, respectively [23]. It
should be noted that based on the Dreicer theory, all particles become runaway in a time period
91
shorter than the collision time if the electric field across the plasma becomes larger than the critical
field.
𝐸(Vm−1) ≈ 3.9 × 10−10 𝑛(cm−3)
𝐾𝑇(eV) (5-8)
The avalanche process is the secondary process that generates runway electrons in a
plasma. In this process, the energetic electrons can produce secondary runaway electrons by
coulomb collisions between two electrons.
The generation of runaway electrons and hard x-rays in a plasma focus based on Dreier
condition yields a good agreement in a low energy DPF [78]. In a plasma focus, the electron
density can reach up to 1019 cm-3, and the plasma temperature can reach to 1 keV during the pinch
phase [78]. The estimated plasma density and temperature in a plasma focus give 40 kVcm-1 as
the critical electric field. Assuming the pinch length is on the order of the anode radius which is
1.5 cm, the calculated critical voltage across the plasma column to produce runaway electrons is
approximately 60 kV. In this experiment, the minimum required plasma voltage, about 25 kV (i.e.
16 kVcm-1 for the electrical field), is required to generate hard x-rays from the device.
It is interesting to calculate the time rate of runaway electron generation in a pinching
plasma by considering the minimum required electric field for hard x-ray radiation from the
plasma. Eq.5-9 shows the time rate of primary runaway generation (based on Dreicer theory) in a
plasma, where νe is the collision frequency of the thermal electron (that is about 109 /s in pinching
plasma), α = E/Ec, where E is the electric field and Ec the critical electric field, and ne the density
of the background electrons [82]. By considering the experimental electric field for hard x-ray
92
generation is at minimum (16 kVcm-1in argon gas, Zi=10) , the estimated time rate of primary
runaway density in plasma is about S D ≃ 10 4 ne (cm -3 s-1)
SD ≃ ne νe 𝛼−
3(𝑍𝑖+1)
16 exp (- 1
4𝛼 -√
(𝑍𝑖+1)
𝛼 ) (5-9)
The time rate of the runaway electron emission based on avalanche process can be
estimated as S a = nr / τav where τav ∝E-1 and is about 10-9 s (based on an electron collision
frequency in pinching plasma), where nr is the density of the runaway electrons,
and Sa ≃ 10 9 × nr (cm -3 s-1) in a plasma focus. In the absence of the loss mechanism, the time rate
of the runaway density can be expressed as dnr /dt = SD + Sa. The solution can be presented by
Eq. 5-10, where t is in the unit of ns.
𝑛𝑟 = 10−5𝑛𝑒(𝑒𝑡 − 1) (5-10)
It is interesting to note that the lifetime of the pinch is on the order of ten nanoseconds, nr
≃10-3 ne. Considering the ratio, 𝑆𝑎/𝑆𝐷 = (𝑒𝑡 − 1), shows that the rate for runaway generation by
the avalanche process becomes greater than the primary process within less than 1 ns in the
pinching plasma, suggesting that the avalanche process plays an important role in the runaway
electron generation in a plasma focus. The generation of hard x-rays has been observed when E/ne
is on the order of 10-16 and nr ≃10-3 ne in a tokamak [82]. Considering that ne ≃ 1016 cm-3 in
tokamaks, the minimum experimental electric field (≈1 V/m) for the hard x-ray generation gives
the same ratio of electric field to the electron density that is observed in a Tokamak although the
two devices are working in two different time scales [83].
93
The comparison between the required electric field to generate hard x-rays in the pinch and
post-pinch phases of the anomalous resistance shows that the required electric field to generate
hard x-rays in the post-pinch phase is less than the required electric field to generate hard x-rays
in the pinch phase. Considering that the elongation speed of plasma column is about 105 m/s,
according to simulation result, the length of the plasma column can increase up to 1 cm in 100 ns.
Comparing the intensity of the plasma voltage at peaks numbered 1 and 2 in Fig. 5.10 (a) and the
time difference of about 120 ns between the two main peaks, the electric field across the plasma
then is about 11 kV/cm in the post-pinch phase.
The comparison between the peaks numbered 1 and 2 in Fig. 5.9 (a) and (b) shows the
approximately the same peak intensity of hard x-ray signals in the post-pinch phase was generated,
and the electric field across the plasma in the post-pinch phase is smaller than the electric field
across the plasma in the pinch phase.
The signature of the anomalous resistance has been found during the post-pinch phase. The
lower required electric field to generate hard x-rays can be interpreted by considering the
anomalous Joule heating and the drop of plasma density in that same phase. Moreover, intense
radiations of soft x-rays have been observed in the post-pinch phase in DPF UofS-I, as can be seen
in Fig. 5.10.
94
Fig. 5.9: (a) tube voltage and discharge current, and (b) discharge current and hard x-ray
signals.
t(µs)
t(µs)
1 2 (b)
t(µs)
V
-Tu
be
(kV
)
(a)
t(µs)
95
The peaks numbered 1 and 2 are correlated with the peaks numbered 1 and 2 in tube
voltages and hard x-rays in Fig. 5.10. This feature can also be interpreted as the effect of the
plasma Joule heating in that same phase.
Fig. 5.10: Soft x-ray signal from DPF UofS-I.
It is interesting to estimate the falling and rising time of the x-ray signal based on the
measured plasma resistance and inductance. The time evolution of the plasma resistance can be
estimated by 𝑅0 [𝑒−𝑡
𝑡2 − 𝑒−𝑡
𝑡1 ] , where 𝑅0 is a constant on the order of 1 Ω [28,45], 𝑡1 is the
characteristic rising time of the anomalous resistance, and 𝑡2 is the characteristic falling time.
The x-rays generated by the anomalous resistance has a fast rising time due to a sudden
occurrence of instabilities. Fig. 5.11 shows the fast rising time of the x-ray signal within 50 ns and
the fast rising time of the second peak of x-ray signal within 10 ns (which is even shorter than the
rising time of the observed during the first peak of the hard x-ray signal). The falling time of the
x-ray signal is about 250 ns.
t(µs)
1 2
96
Fig. 5.11: Rising time and falling time of x-ray signal.
The observed hard x-ray falling time can be estimated by t2= LPlasma /RAnomalous
(a characteristic time to transfer energy from the stored magnetic energy into the plasma through
anomalous resistance). Experimentally, the falling time of the second x-ray peak is on the order of
hundreds of nanoseconds. By considering the plasma inductance on the order of 10-8 nH, as
discussed earlier, and the plasma resistance in the order of 0.1 Ω, the falling time t2 is then around
10-7 s, corresponding to the observed falling time of the x-ray signal.
97
5.5 DPF Device with Semi-Active Electrodes (DPF-SAE)
Fusion research on a DPF device was put on hold due to the saturation of the neutron yield
with increasing stored energy of a DPF device. Moreover, operating a high-energy DPF device
causes many technical problems due to the impurities and instability effects that prevents
formation of the focusing plasma. To solve these problems, a staged-plasma focus is implemented
by using two plasma focus devices that can be placed in front of each other, and combined through
a mid-plane disk. Furthermore, it has been suggested to increase the neutron yield beyond the
established scaling law for a single-gun DPF device [84]. However, the main problem in these
configurations is the timing of the focusing event in each of the DPF device. It should be noted
that the delay time between the two focusing events in the two DPF devices with the same
operating conditions is mainly caused by the delay time between the switching of spark gaps, and
the delay time between the occurrence of breakdown in the two discharge tubes. The time
difference of two focusing events can be improved in a hypocycloidal pinch [85], which consists
of two cathode disks with a common anode disk. In this configuration, two focusing events could
generate charged particles in the opposite directions.
In this Section a new configuration of a DPF device built will be introduced. In the first
part, the technical details of an SAE-type plasma focus will be presented. In the second part, the
electrical discharge signals of the device will be shown and analyzed. In the third part, the
generation of charged particles and x-rays from the device will be presented. The results
corresponding to the charged particles and x-ray radiations from a DPF-SAE and the DPF UofS-I
will be compared.
98
5.5.1 Configuration of DPF-SAE
The cross view of an SAE-type plasma focus is shown in Fig. 5.12. The system is the same
as DPF UofS-I except the electrode head. In the SAE-type plasma focus, three electrodes that are
isolated by insulators from each other are used to form two discharge tubes. The inner electrode
(1) is connected to the capacitor bank through a spark gap switch. The middle electrode (2) is
isolated electrically from the capacitor bank. The electrode 2 is termed the semi-active electrode
in this thesis. The outer electrode (3) is connected to the capacitor bank through the ground
connection. Two current layers can be formed between the two discharge tubes after the discharge
of a single capacitor bank. The current layers are then accelerated by the J× B force in the axial
and radial phases. The electrodes 1 and 2 are cylindrical copper electrodes, and electrode 3 is made
of 12 copper rods. Fig. 5.13 shows the equivalent circuit of an SAE-type plasma focus.
99
Fig. 5.12: Cross view of DPF-SAE, The red color shows the copper and blue color shows
the insulator.
Cross view
2
3
1
100
Fig. 5.13: The equivalent circuit of SAE type plasma focus.
Fig. 5.14 is a picture of the DPF-SAE assembly. To optimize the geometry of the three
electrodes, the Lee model has been used for two independent DFP devices. Assuming that DPF
UofS-I is a combination of electrodes 1 and 2, the Lee model gives a length of 0.5 cm for the radius
of electrode 1, and 1.5 cm for the radius of electrode 2, to achieve the good focusing conditions in
a deuterium gas at a pressure of 3 Torr. Assuming DPF-II as a combination of electrodes number
2 and number 3, Lee model gives 1.5cm for the radius of electrode number 2 and 5 cm for the
radius of electrode number 3 to achieve best focusing conditions in Deuterium gas at 3 Torr. Figure
5.15 shows the focusing effect in the DPF-I and DPF-II, based on simulation results. The Lee
model gives a rough estimation of the dimensions of three electrodes. A new code has to be
developed to optimize the geometry of the electrodes for the SAE-type dense plasma focus devices.
The timing of the two focusing events can be synchronized by: (a) a proper length of
electrodes, and (b) a proper length of insulators between the electrodes. It should be noted that in
an SAE-type plasma focus, the spark gap and the transmission line are common between the
C0
S
H.V
L1-2(t), r1-2(t) L2-3(t), r2--3(t)
L0, rO
Electrodes 1&2 Electrodes 2&3
101
electrodes. The breakdown between electrodes happens at the same time as they are in series. Table
5.2 shows the parameters predicted by the Lee model for an SAE-type plasma focus device.
Fig. 5.14: Experimental set-up of electrodes in SAE type plasma focus.
102
Fig. 5.15: Simulated discharge current and tube voltage using Lee model separately for
DPF-I and DPF-II.
103
Table 5.2: Parameters predicted by Lee model for SAE type plasma focus.
Value Parameter
F 4.9 Capacitance (C 0)
150 nH Inductance e (L0)
28 kV Charging voltage (V0)
kJ 2 Stored energy (E 0)
140 kA Peak discharge current
cm 0.5 Electrode number 1 radius (a)
cm 1.5 Electrode number 2 radius(b)
5 cm Electrode number3 radius(c)
15 cm Electrode number 1ength (L-1)
4 cm Electrode number 2 length (L-2)
4 cm Electrode number 3 length (L-3)
3 Torr Operating pressure
An SAE type-plasma focus has been operated in hydrogen, nitrogen, and argon gas. To
measure the discharge current and voltage, a current probe and two high voltage probes have been
used. The connection of the probes can be seen in Fig. 5.16. V1 and V2 are the measured voltages
of the central electrode 1 and the semi-active electrode 2 with the grounded cathode 3. The current
probe measures the current to the anode. The semi-active electrode is insulated from from the
anode and cathode and is connected to the anode and cathode through the plasma.
104
Fig. 5.16: Connection of current probe and two high voltage probes.
105
The measured waveforms of the discharge current and tube voltages in an SAE-type
plasma focus operating in argon gas are shown in Fig. 5.17. The first drop of current shows the
focusing effect in DFP-I, and the second current drop shows focusing effect in DPF-II. Figure 5.17
shows that the second focusing event starts right after the first focusing. The operating pressure
can vary the timing and the number of focusing events, as can be seen in Fig. 5.18. To measure
the voltage across the plasma between each pair of electrodes, circuit analyses based on the
waveforms of the discharge current and high voltages are necessary.
Fig. 5.17: Discharge current and tube show two focusing events occur almost at the same
time. The operating pressure P≈ 0.1 Torr, argon gas.
V1 (
kV
), V
2(k
V),
Cu
rren
t(k
A)
t(µs)
106
Fig. 5.18: Discharge current and tube voltages show three focusing events occur at
different time. The operating pressure P ≈0.2 Torr, argon gas.
5.5.2 Discharge Circuit Analyses in DPF-SAE
The voltage across the discharge electrodes (V1 = Vtube) can be expressed by Eq. 5-11,
1 0 0 0
0
( ) ( ) ( )Tube c c c
d dI Idt dIV V L t f I L t f r t f I V R I L
dt dt c dt (5-11)
where fc is the fraction of the discharge current that passes through the plasma, V0 is the charging
voltage of the capacitor, C0 is the capacitance. L (t) = L1-2 (t) + L2-3 (t), and r(t) = r1-2 (t) + r2-3 (t).
L1-2 (t) and r1-2 (t) refer to the plasma inductance and resistance that is formed between electrodes
1 and 2. L2-3 (t) and r2-3 (t) refer to plasma inductance and resistance between electrode number 2
and 3.
V1 (
kV
),V
2 (
kV
), C
urr
en
t(k
A)
t(µs)
107
The tube voltage can be calculated using the discharge current and anode voltage, based
on Eq. 5-12,
𝑉𝑇𝑢𝑏𝑒(𝑡) ≃ 𝑉𝐴(𝑡) − 𝐿0𝑑𝐼
𝑑𝑡 (5-12)
where 𝑉𝐴(𝑡) is the anode voltage, 𝐼(𝑡) is the discharge current, and 𝐿0 is the inductance of the
discharge circuit from the voltage probe to the discharge electrodes. Considering that 𝐿0 = 70 nH
in the experiment, the measured tube voltages can be seen in Fig. 5.19, where three peaks of
voltages correspond to the three focusing events. The voltage between the electrodes 2 and 3 (i.e.
V2) and the difference between V1 and V2, can be seen in Fig. 5.20.
The voltage corresponding to V2 shows one peak, while the voltage difference V1 – V2
shows two main peaks. The last peak in the V1 – V2 signal is correlated with the peak voltage in V2.
A voltage drop can be observed in the V2 signal when the peak of the voltage appears in V1.
Increasing the impedance of the plasma between the electrodes 1 and 2 causes a drop of voltage in
V2 since the voltage V1 is divided between the electrodes 1 and 2 and the electrodes 2 and 3 (i.e.
V2). Increasing the plasma impedance between electrodes 2 and 3 causes a peak voltage in the V2
signal.
108
Fig. 5.19: The waveforms of the tube voltage and discharge current in DPF-SAE. P≈0.1
Torr, argon gas.
Fig. 5.20: The voltage on electrodes number 2 and the difference between V1 and V2.
Three peaks in V1 –V2 shows an increase of plasma impedance between the electrodes 1
and 2 in three different stages. The single peak in the V2 signal shows an increase of plasma
impedance between electrodes 2 and 3 in a single stage. The first peak in V1 –V2 corresponds to the
109
first focusing event between electrodes 1 and 2. The second peak in V1 –V2 appears at the starting
time of the main peak in V2. The third peak in V1 –V2 is correlated with the first peak in V2. The
duration of this peak is shorter than the duration of the main peak in V2. The main peak in V2
corresponds to the focusing of the plasma between electrodes 2 and 3. It should be noted that the
nature of the second and third peaks in V1 –V2 signals is not yet clear.
5.5.3 Charged Particle Emission and X-ray Radiation from the DPF-
SAE Device
Fig. 5.21 shows the ion beam and hard x-ray signals from the DPF-SAE device together
with the current signal. The corresponding V1 and V2 – V1 signals were presented in Fig. 5.20. There
are three peaks in the ion beam signal and two peaks in hard x-ray signal. The energy of the hard
x-ray exceeds 100 keV based on the filter used. The duration of the hard x-ray radiation can reach
up to 500 ns, and the duration of the ion beam can reach up to 1μs. It is interesting to note that the
second peak in the hard x-ray signal appears after the second peak in ion beam signal. Figure 5.22
shows the waveforms of ion beam emissions and hard x-ray radiations from another DPF-SAE
discharge which resulted in three peaks in both signals (the number of peaks vary between two
and three peaks in the experiment). The first peak of ion beam signal is correlated with the first
focusing event of the plasma between electrodes 1 and 2 in the SAE-type plasma focus. The second
peak of the ion beam appears before the second focusing event. The third peak of ion beam signal
is correlated with the second focusing event of the plasma between electrodes 2 and 3.
110
Fig. 5.21: Three periods of ion beam emissions and hard x-ray radiations from DPF-SAE.
Fig. 5.22: Three periods of ion beam emissions and hard x-ray radiation from DPF-SAE.
1
2 3
t(µs)
1 2
3
t(µs)
111
Based on the strong radiations of hard x-rays and energetic ions, significant radiation of
the electron beams, soft x-rays, and neutrons from the device, are also expected. Anomalous
resistance, rapid change in plasma inductance, and magnetic reconnection of the two current
layers, might be the possible mechanisms of charged particle and x-ray radiations from the plasma.
The occurrence of two focusing events in a proper time frame can enhance the duration of
charged particle emissions and x-ray radiations from the DPF-SAE device. Further experiments
are needed to synchronize the two focusing events by changing the geometry of electrodes, the
working pressure, and the insulator length.
5.5.4 Comparison of Ion Emissions and X-ray Radiations between
DPF-SAE and DPF UofS-I Devices
The DPF-SAE and the DPF UofS-I devices were operated under the same conditions, e.g.,
with the same stored energy (2 kJ). Figure 5.23 shows comparison of the ion beam signals from
the two different DPF configurations. The longer duration of the ion beam with a higher intensity
in the SAE-type plasma focus can be explained by the longer duration of the current drop and the
tube voltage.
112
Fig. 5.23: Comparison of ion beam garnered from DPF-SAE and DPF-UofS-I devices.
Figure 5.24 shows the comparison between the hard x-ray signals in the DPF-SAE and the
DPF UofS-I devices. The duration of hard x-ray signal that is generated by the DPF-SAE is about
500 ns, while the hard x-ray signal from the DPF UofS-I lasts only up to 300 ns. A significant
increase in hard x-ray intensity generated from the DPF-SAE device compared to DPF UofS-I
device shows a more effective operation in the DPF-SAE as a new type of hard x-ray based on
DPF. Intense radiations of hard x-rays show strong bombardment on electrodes due to energetic
electrons. The first peak in the hard x-ray signal is correlated with the first focusing event of the
plasma between electrodes 1 and 2, while the second peak of the ion beam appears before the
second focusing event. The third peak of the hard x-ray signal is correlated with second focusing
event occurring between electrodes 2 and 3 in the SAE-type plasma focus.
t(µs)
113
The results corresponding to the tube voltage, ion beam emission, and hard x-ray radiation
from the DPF-SAE device, compared to DPF UofS-I device, show a higher injected energy into
the plasma, and a better operation overall as an ion and x-ray source. The study of soft x-ray
radiations and neutron emissions from the SAE-type plasma focus are some of the interesting
future research that can be implemented.
Fig. 5.24: The comparison of hard x-ray signals in DPF-SAE and DPF UofS-I device.
DPF-SAE
DPF-UofS-I
t(µs)
t(µs)
114
Chapter 6
SUMMARY
6.1 Summary
In this thesis, a brief introduction on magnetic pinch devices has been presented. The Lee
model, a well-known model in simulating the phenomena in a focusing plasma, has been
introduced. The design and optimization of a low energy plasma focus in enhancing charged
particle emissions and x-ray radiation from the device have been discussed. The design and the
technical details of a dense plasma focus have been explained. The Lee model has been used to
predict the optimum ion current emission from the device. Several detectors, such as a Faraday
cup, a soft x-ray detector, a hard x-ray detector, and an electron beam detector, have been
developed, and the technical details have been introduced.
A series of experiments have been performed on the DPF UofS-I. Different regimes of
focusing have been observed in three operating gases (i.e. hydrogen, nitrogen, and argon). The
results show that the duration of the current drop and the injected energy into the plasma increased
significantly in the argon gas compared to the hydrogen gas. For all three operating gasses, a circuit
analysis has been performed to study the plasma voltage, the plasma inductance, and the injected
energy into the plasma. The plasma voltage, plasma inductance, and injected energy into plasma
are maximized in DPF UofS-I in argon gas. The maximum duration of the current drop has also
115
been observed when argon is used as the operating gas. Based on simulation results, the radiation-
enhanced-compression of the plasma and the anomalous resistance have been adopted as the
mechanism to explain the measured plasma inductance and plasma voltage in a focused argon
plasma. Circuit analysis has been performed to verify the simulation results. The measured and
simulated inductance in the argon plasma agree with each other. The measured plasma resistance
can also verify the simulation results regarding the occurrences of the anomalous resistance in the
post-pinch phase. The measured anomalous resistance reaches up to 0.2ohms after the pinch phase.
This result also confirms the inductive or resistive nature of the current drop and the peak of the
plasma voltage after the pinch phase.
A significant contribution of the plasma resistance, in the energy consumption by a
pinching plasma in argon gas, has been related to the required condition of LHDI in plasmas as
one of the main instabilities that makes the plasma resistance anomalous. A drop of temperature
in the post pinch-phase decreases the effective charge of the ion in that phase. Decreasing the
effective charge of the heavy ions and plasma density in the post-pinch phase is the favourable
condition for LHDI in plasmas. The occurrence of the anomalous Joule heating by the onset of the
anomalous resistivity in the post pinch phase can also terminate the radiation-enhanced
compression of the plasma. The energy spectrum of the ions generated in three operating gases
show that the ion energy depends on the plasma voltage and the effective charge of the ion.
Increasing the atomic number of the gas could enhance the energy of generated ions. The results
corresponding to the ion current density are in agreement with the simulation results based on the
Lee model.
116
The required experimental E-field across the plasma to generate significant runaway
electrons and hard x-rays during the pinch phase and the phase with anomalous resistance, have
been studied. The results show the significant generation of charged particles and hard x-rays at
smaller E-field across the plasma column in the phase of anomalous resistances compared to the
pinch phase. The signature of the anomalous resistance has been found in the post-pinch phase.
The anomalous plasma heating process may enhance generation of runaway charged particle
density in the plasma during the phase of anomalous resistance due to reduced Dreicer field. This
result shows that plasma heating can enhance the efficiency of the plasma focus as a charged
particle source.
Runaway electron generation in plasma focus devices and tokamaks have been compared
by considering the primary (Dreicer theory) and secondary (Avalanche theory) runaway electron
production. It has been found that the runaway electron generation conditions in tokamaks and
plasma focus devices are consistent in terms of ratios between 𝐸 and 𝑛𝑒 , and between 𝑛𝑟 and 𝑛𝑒.
This is one of the important results that could open a new insight regarding runaway electron
generation in large fusion reactors.
Further enhancement of the plasma focus efficiency as a charged particle source has been
implemented through a new configuration of the discharge electrodes. The SAE-type plasma
focus, with a combination of three electrodes to produce several focusing events, has been
developed for the first time. Analysis of the discharge circuit of the device shows the occurrence
of the three focusing events. The preliminary results corresponding to ion beam emissions and
hard x-ray radiations from a DPF-SAE device are very promising. The long duration and high
117
intensity of ion beam and hard x-ray signals from a DPF-SAE show a higher efficiency of this
device as a charged particle source for medical and industrial applications.
6.2 Suggested Future Research
Runaway electron generation by considering loss process in a plasma focus is one the
interesting research areas that can be pursued on DPF UofS-I. Beside, LHDI instability in pinching
plasma can be studied by using ultrafast oscilloscope ( 10 GHZ) to see the effects of this instability
on occurrence of anomalous resistance in plasma. Plasma temperature measurement in post pinch
phase of anomalous resistances also can provide extra evidence of anomalous heating in that phase.
Modeling of pinching plasma in a DPF-SAE device based on electrodynamics and
thermodynamics of the plasma can be an interesting theoretical research area to predict the
behavior of plasma in the new plasma focus configuration. Optimization of DPF-SAE device based
on geometry of discharge electrodes and insulator length can be a matter of new research to
enhance beam generation from the device.
Mechanisms of charged particle and x-ray generation in a SAE type plasma focus, energy
spectrum of ion beam, electron beam and x-rays generated from an SAE type plasma focus, and
plasma temperature measurement during focusing phase in an SAE type plasma focus can also
give an understanding about formation and interaction of several pinch accelerators in a plasma
focus device.
118
Neutron emission from a SAE type plasma focus is another subject of research that can
be studied after some lab safety considerations. Measuring the energy and number of neutrons
would also give some information about the interaction of several pinch regions in the plasma.
Operation of medium and high stored energy SAE type plasma focus, and developing an
SAE type plasma focus for medical and industrial applications also are suggested future work that
can be planned based on the outcome of this thesis.
119
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APPENDIX
1-Electrodes assembly:
